Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7625195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×D16 |
| commutator, Frattini, cosocle | C16 | C25 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1320 conjugacy classes of subgroups; only the 528 normal subgroups are shown.
This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 528 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 — 5 generators on 128 points:
(1,2)(3,15)(4,10)(5,25)(6,31)(7,35)(8,11)(9,17)(12,28)(13,27)(14,33)(16,52)(18,62)(19,65)(20,53)(21,41)(22,69)(23,74)(24,76)(26,49)(29,85)(30,51)(32,47)(34,87)(36,54)(37,56)(38,61)(39,81)(40,80)(42,72)(43,71)(44,68)(45,59)(46,63)(48,101)(50,60)(55,104)(57,110)(58,112)(64,102)(66,103)(67,115)(70,97)(73,122)(75,95)(77,105)(78,83)(79,84)(82,100)(86,92)(88,99)(89,106)(90,120)(91,119)(93,108)(94,111)(96,127)(98,109)(107,128)(113,126)(114,118)(116,125)(117,121)(123,124);(1,3,8,20)(2,9,11,38)(4,16,21,55)(5,26,30,82)(6,32,34,88)(7,36,33,19)(10,37,41,89)(12,47,45,99)(13,49,46,100)(14,50,35,40)(15,51,53,25)(17,59,61,28)(18,63,64,27)(22,70,66,113)(23,75,67,116)(24,77,68,58)(29,86,84,81)(31,39,87,92)(42,95,93,125)(43,97,94,126)(44,98,76,91)(48,102,78,62)(52,103,104,69)(54,79,65,85)(56,108,106,72)(57,111,107,71)(60,83,80,101)(73,123,114,120)(74,90,115,124)(96,128,117,110)(105,118,112,122)(109,121,119,127);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,21)(9,37)(11,41)(12,42)(13,43)(14,44)(15,52)(17,56)(18,57)(19,58)(20,55)(25,69)(26,70)(27,71)(28,72)(29,73)(30,66)(31,74)(32,75)(33,68)(34,67)(35,76)(36,77)(38,89)(39,90)(40,91)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,103)(53,104)(54,105)(59,108)(60,109)(61,106)(62,110)(63,111)(64,107)(65,112)(78,117)(79,118)(80,119)(81,120)(82,113)(83,121)(84,114)(85,122)(86,123)(87,115)(88,116)(92,124)(99,125)(100,126)(101,127)(102,128);(4,21)(10,41)(16,55)(22,66)(23,67)(24,68)(37,89)(42,93)(43,94)(44,76)(52,104)(56,106)(57,107)(58,77)(69,103)(70,113)(71,111)(72,108)(73,114)(74,115)(75,116)(90,124)(91,98)(95,125)(96,117)(97,126)(105,112)(109,119)(110,128)(118,122)(120,123)(121,127);(2,11)(9,38)(10,41)(12,45)(13,46)(14,35)(15,53)(25,51)(31,87)(37,89)(39,92)(40,50)(42,93)(43,94)(44,76)(47,99)(48,78)(49,100)(52,104)(54,65)(62,102)(69,103)(74,115)(79,85)(90,124)(91,98)(95,125)(96,117)(97,126)(105,112)(110,128)(118,122)Polycyclic presentation:
2^9|b^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,f^b=fg,g^b=gh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532392);