Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393928 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14715 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,10)(5,25)(6,13)(7,12)(8,14)(9,18)(11,27)(16,42)(17,45)(19,33)(20,32)(21,30)(22,51)(23,37)(24,38)(26,40)(28,39)(29,36)(31,47)(34,44)(35,52)(41,48)(43,56)(46,55)(49,57)(50,58)(53,59)(54,60)(61,62)(63,64);(1,3)(2,9)(4,21)(5,17)(6,20)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,23)(22,50)(24,49)(25,55)(26,48)(27,46)(28,47)(29,44)(30,37)(35,58)(36,57)(38,42)(39,41)(40,45)(43,53)(51,56)(52,63)(54,62)(59,61)(60,64);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,35)(12,37)(13,36)(14,38)(15,34)(17,43)(18,49)(19,44)(20,21)(25,51)(26,53)(27,52)(28,54)(31,56)(32,57)(33,42)(39,60)(40,59)(41,58)(45,64)(46,50)(47,63)(48,62)(55,61);(1,5)(2,11)(3,17)(4,22)(6,27)(7,26)(8,28)(9,31)(10,35)(12,25)(13,39)(14,40)(15,41)(16,43)(18,48)(19,47)(20,46)(21,50)(23,53)(24,54)(29,52)(30,56)(32,45)(33,55)(34,58)(36,60)(37,51)(38,59)(42,61)(44,63)(49,62)(57,64)Polycyclic presentation:
2^9|e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301124);