Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513188 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #2170 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,17)(3,18,24,72)(4,25,29,64)(5,30,37,94)(6,16,8,14)(7,40,43,65)(10,45,50,21)(11,38,53,27)(12,54,58,42)(13,59,63,35)(15,39,67,33)(19,47,77,113)(20,55,81,119)(22,83,44,71)(23,84,85,124)(26,56,88,121)(28,57,90,52)(31,66,97,125)(32,86,100,123)(34,93,36,92)(41,60,105,122)(46,108,110,79)(48,82,68,96)(49,103,115,95)(51,104,116,98)(61,102,62,101)(69,76,111,74)(70,106,109,73)(75,114,107,128)(78,117,80,118)(87,91,99,120)(89,112,126,127);(1,3)(2,10)(4,19)(5,31)(6,22)(7,41)(8,44)(9,24)(11,46)(12,55)(13,60)(14,48)(15,66)(16,68)(17,50)(18,30)(20,26)(21,35)(23,33)(25,70)(27,74)(28,89)(29,77)(32,73)(34,95)(36,103)(37,97)(38,76)(39,85)(40,49)(42,75)(43,105)(45,59)(47,56)(51,114)(52,117)(53,110)(54,107)(57,118)(58,119)(61,84)(62,124)(63,122)(64,109)(65,115)(67,125)(69,91)(71,92)(72,94)(78,87)(79,98)(80,99)(81,88)(82,102)(83,93)(86,112)(90,126)(96,101)(100,106)(104,108)(111,120)(113,121)(116,128)(123,127);(1,4)(2,11)(3,19)(5,26)(6,38)(7,28)(8,27)(9,29)(10,46)(12,40)(13,51)(14,64)(15,52)(16,25)(17,53)(18,69)(20,31)(21,73)(22,76)(23,75)(24,77)(30,91)(32,35)(33,42)(34,99)(36,87)(37,88)(39,54)(41,89)(43,90)(44,74)(45,106)(47,83)(48,109)(49,55)(50,110)(56,93)(57,67)(58,65)(59,100)(60,114)(61,123)(62,86)(63,116)(66,117)(68,70)(71,113)(72,111)(78,103)(79,96)(80,95)(81,97)(82,108)(84,127)(85,107)(92,121)(94,120)(98,101)(102,104)(105,126)(112,124)(115,119)(118,125)(122,128);(1,4)(2,12)(3,20)(5,32)(6,27)(7,42)(8,38)(9,29)(10,47)(11,40)(13,51)(14,57)(15,25)(16,52)(17,58)(18,70)(19,31)(21,75)(22,78)(23,73)(24,81)(26,35)(28,33)(30,56)(34,98)(36,104)(37,100)(39,90)(41,79)(43,54)(44,80)(45,107)(46,83)(48,111)(49,114)(50,113)(53,65)(55,60)(59,88)(61,86)(62,123)(63,116)(64,67)(66,112)(68,69)(71,110)(72,109)(74,95)(76,103)(77,97)(82,126)(84,118)(85,106)(87,102)(89,96)(91,93)(92,120)(94,121)(99,101)(105,108)(115,128)(117,124)(119,122)(125,127)Polycyclic presentation:
2^9|a^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gh,h^a=hi,d^b=de,e^b=eg,g^b=gh,h^b=hi,f^c=fi,e^d=eg,g^d=gh,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420385);