Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417618 |
| 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 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,8,15,28)(4,9,18,31)(5,20,21,34)(6,23,25,11)(10,24,33,22)(13,35,37,26)(14,39,40,55)(16,43,42,29)(17,45,47,58)(19,50,49,32)(27,44,54,41)(30,51,57,48)(36,60,46,64)(38,53,62,59)(52,61,56,63);(1,3)(2,8)(4,13)(5,14)(6,16)(7,15)(9,26)(10,27)(11,29)(12,28)(17,46)(18,37)(19,38)(20,39)(21,40)(22,41)(23,43)(24,44)(25,42)(30,56)(31,35)(32,53)(33,54)(34,55)(36,47)(45,60)(48,63)(49,62)(50,59)(51,61)(52,57)(58,64);(1,4)(2,9)(3,13)(5,17)(6,19)(7,18)(8,26)(10,30)(11,32)(12,31)(14,36)(15,37)(16,38)(20,45)(21,47)(22,48)(23,50)(24,51)(25,49)(27,52)(28,35)(29,53)(33,57)(34,58)(39,64)(40,46)(41,61)(42,62)(43,59)(44,63)(54,56)(55,60);(1,5)(2,10)(3,14)(4,17)(6,24)(7,21)(8,27)(9,30)(11,34)(12,33)(13,36)(15,40)(16,44)(18,47)(19,51)(20,23)(22,25)(26,52)(28,54)(29,55)(31,57)(32,58)(35,56)(37,46)(38,63)(39,43)(41,42)(45,50)(48,49)(53,60)(59,64)(61,62)Polycyclic presentation:
2^9|a^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286082, #72286176, … |
| maximal normal subgroup of | #72285925, #72286082, #72286176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324815);