Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419790 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1543 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)(3,15,19,34)(4,10)(5,25,28,41)(6,12)(7,13)(8,14)(9,18,33,29)(11,27,40,30)(16,43,46,35)(17,32)(20,49,45,31)(21,51,53,55)(22,37)(23,42)(24,38)(26,39)(36,54,62,52)(44,57)(47,58)(48,59)(50,63)(56,60)(61,64);(1,3)(2,9)(4,20)(5,26)(6,29)(7,18)(8,19)(10,35)(11,39)(12,34)(13,15)(14,33)(16,23)(17,27)(21,44)(22,46)(24,45)(25,32)(28,47)(30,48)(31,37)(36,61)(38,43)(40,58)(41,59)(42,49)(50,52)(51,63)(53,64)(54,56)(55,60)(57,62);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,31)(11,36)(12,37)(13,42)(14,38)(15,35)(17,44)(18,45)(19,46)(20,29)(25,55)(26,56)(27,52)(28,53)(30,54)(32,57)(33,49)(34,43)(39,60)(40,62)(41,51)(47,50)(48,64)(58,63)(59,61);(1,5,6,27,8,28,7,30)(2,11,12,25,14,40,13,41)(3,17,18,26,19,48,29,47)(4,21,22,52,24,53,23,54)(9,32,15,39,33,59,34,58)(10,36,37,55,38,62,42,51)(16,44,45,56,46,64,20,50)(31,57,35,60,49,61,43,63)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326986);