Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #513316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 ◁ #2258 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,10)(4,9)(5,26)(6,30)(7,33)(8,15)(11,21)(12,22)(13,27)(14,31)(16,39)(17,34)(18,60)(19,63)(20,64)(23,49)(24,51)(25,38)(28,80)(29,78)(32,82)(35,66)(36,65)(37,73)(40,56)(41,57)(42,61)(43,55)(44,72)(45,75)(46,68)(47,81)(48,71)(50,79)(52,84)(53,104)(54,107)(58,97)(59,98)(62,116)(67,83)(69,77)(70,76)(74,99)(85,109)(86,108)(87,103)(88,121)(89,120)(90,119)(91,100)(92,101)(93,105)(94,112)(95,113)(96,117)(102,125)(106,115)(110,118)(111,114)(122,127)(123,126)(124,128);(2,9)(4,21)(5,22)(6,27)(7,31)(10,34)(11,12)(13,49)(14,51)(15,38)(17,56)(18,57)(19,61)(20,55)(23,72)(24,75)(25,68)(26,35)(28,81)(29,71)(30,36)(32,79)(33,37)(39,84)(40,41)(42,97)(43,98)(44,77)(45,76)(46,48)(47,83)(50,99)(52,100)(53,101)(54,105)(58,112)(59,113)(60,85)(62,117)(63,86)(64,87)(65,66)(67,73)(69,70)(74,121)(78,89)(80,88)(82,90)(91,92)(93,125)(94,115)(95,114)(96,118)(102,128)(103,110)(104,122)(106,111)(107,123)(108,109)(116,124)(119,120)(126,127);(1,3)(2,10)(4,17)(5,18)(6,19)(7,20)(8,16)(9,34)(11,40)(12,41)(13,42)(14,43)(15,39)(21,56)(22,57)(23,58)(24,59)(25,52)(26,60)(27,61)(28,62)(29,53)(30,63)(31,55)(32,54)(33,64)(35,85)(36,86)(37,87)(38,84)(44,94)(45,95)(46,91)(47,96)(48,92)(49,97)(50,93)(51,98)(65,108)(66,109)(67,110)(68,100)(69,106)(70,111)(71,101)(72,112)(73,103)(74,102)(75,113)(76,114)(77,115)(78,104)(79,105)(80,116)(81,117)(82,107)(83,118)(88,124)(89,122)(90,123)(99,125)(119,126)(120,127)(121,128);(3,16)(10,39)(17,52)(18,53)(19,54)(20,55)(34,84)(40,91)(41,92)(42,93)(43,64)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(85,122)(86,123)(87,98)(94,124)(95,118)(96,114)(97,125)(108,126)(109,127)(110,113)(111,117)(112,128)(115,116)Polycyclic presentation:
2^9|e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=be,e^a=ef,f^a=fg,g^a=gh,h^a=hi,e^b=ef,f^b=fg,g^b=gh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321377 |
| maximal normal subgroup of | #72321377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420512);