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)(3,18,24,73)(4,25,29,64)(5,30,37,69)(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)(19,47,78,114)(20,55,83,120)(22,85,44,72)(23,86,87,124)(26,56,79,111)(28,57,91,52)(31,66,97,125)(32,88,100,123)(34,94,36,93)(41,60,105,122)(46,108,110,81)(48,84,68,96)(49,103,116,95)(51,104,117,98)(61,102,62,101)(70,77,112,75)(71,106,109,74)(76,115,107,128)(80,118,82,119)(89,92,99,121)(90,113,126,127);(1,3,9,24)(2,10,17,50)(4,19,29,78)(5,31,37,97)(6,22,8,44)(7,41,43,105)(11,46,53,110)(12,55,58,120)(13,60,63,122)(14,48,16,68)(15,66,67,125)(18,69,73,30)(20,79,83,26)(21,59,45,35)(23,39,87,33)(25,71,64,109)(27,75,38,77)(28,90,91,126)(32,74,100,106)(34,95,36,103)(40,49,65,116)(42,76,54,107)(47,111,114,56)(51,115,117,128)(52,118,57,119)(61,86,62,124)(70,121,112,92)(72,94,85,93)(80,99,82,89)(81,104,108,98)(84,101,96,102)(88,113,123,127);(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)(18,70)(20,31)(21,74)(22,77)(23,76)(24,78)(30,92)(32,35)(33,42)(34,99)(36,89)(37,79)(39,54)(41,90)(43,91)(44,75)(45,106)(47,85)(48,109)(49,55)(50,110)(56,94)(57,67)(58,65)(59,100)(60,115)(61,123)(62,88)(63,117)(66,118)(68,71)(69,121)(72,114)(73,112)(80,103)(81,96)(82,95)(83,97)(84,108)(86,127)(87,107)(93,111)(98,101)(102,104)(105,126)(113,124)(116,120)(119,125)(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)(18,71)(19,31)(21,76)(22,80)(23,74)(24,83)(26,35)(28,33)(30,56)(34,98)(36,104)(37,100)(39,91)(41,81)(43,54)(44,82)(45,107)(46,85)(48,112)(49,115)(50,114)(53,65)(55,60)(59,79)(61,88)(62,123)(63,117)(64,67)(66,113)(68,70)(69,111)(72,110)(73,109)(75,95)(77,103)(78,97)(84,126)(86,119)(87,106)(89,102)(90,96)(92,94)(93,121)(99,101)(105,108)(116,128)(118,124)(120,122)(125,127)Polycyclic presentation:
2^9|a^2=i,b^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, 420387);