Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2305 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)(4,20)(5,25)(6,14)(7,12)(8,13)(9,18)(10,23)(11,28)(16,31)(17,32)(19,41)(21,52)(22,37)(24,36)(26,43)(27,40)(29,39)(30,33)(34,47)(35,56)(38,49)(42,46)(44,61)(45,50)(48,57)(51,54)(53,60)(55,58)(59,63)(62,64);(1,3)(2,9)(4,16)(5,26)(6,30)(7,18)(8,19)(10,34)(11,38)(12,41)(13,33)(14,15)(17,27)(20,50)(21,53)(22,46)(23,45)(24,47)(25,57)(28,48)(29,49)(31,37)(32,39)(35,61)(36,42)(40,43)(44,54)(51,60)(52,62)(55,64)(56,63)(58,59);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,31)(11,35)(12,36)(13,37)(14,20)(15,42)(17,44)(18,45)(19,47)(25,58)(26,53)(27,54)(28,55)(29,56)(30,46)(32,59)(33,34)(38,62)(39,51)(40,52)(41,50)(43,61)(48,64)(49,63)(57,60);(1,5,8,29)(2,11,13,40)(3,17,19,48)(4,21,24,56)(6,28,7,27)(9,32,33,57)(10,35,37,52)(12,25,14,39)(15,43,41,38)(16,44,47,64)(18,49,30,26)(20,51,36,58)(22,55,23,54)(31,59,34,60)(42,61,50,62)(45,63,46,53)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318900, #72318935, #72319003, … |
| maximal normal subgroup of | #72318900, #72318935, #72319003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324933);