Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623326 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,20)(11,21,22,47)(12,17,26,40)(13,29,33,48)(14,34,31,19)(16,36,37,24)(18,43,45,32)(23,49,50,69)(25,53,35,44)(27,56,54,70)(28,59,55,42)(30,46,61,39)(38,62,63,51)(41,66,65,58)(52,71,60,67)(57,68,72,64);(1,3)(2,5)(4,8)(6,7)(11,22)(16,37)(21,47)(23,50)(24,36)(25,35)(30,61)(38,63)(39,46)(44,53)(49,69)(51,62)(52,60)(57,72)(64,68)(67,71);(3,8)(5,7)(9,11,15,22)(10,16,20,37)(12,23,26,50)(13,30,33,61)(14,35,31,25)(17,38,40,63)(18,44,45,53)(19,46,34,39)(21,48,47,29)(24,43,36,32)(27,57,54,72)(28,60,55,52)(41,67,65,71)(42,68,59,64)(49,70,69,56)(51,66,62,58);(2,6)(5,7)(9,12)(10,17)(11,23)(13,27)(14,28)(15,26)(16,38)(18,41)(19,42)(20,40)(21,49)(22,50)(24,51)(25,52)(29,56)(30,57)(31,55)(32,58)(33,54)(34,59)(35,60)(36,62)(37,63)(39,64)(43,66)(44,67)(45,65)(46,68)(47,69)(48,70)(53,71)(61,72);(12,26)(17,40)(23,50)(27,54)(28,55)(38,63)(41,65)(42,59)(49,69)(51,62)(52,60)(56,70)(57,72)(58,66)(64,68)(67,71)Polycyclic presentation:
2^9|a^2=i,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cfi,g^c=gh,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530524);