Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418476 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,34)(4,10,24,38)(5,25,29,11)(6,13,7,12)(9,20,33,18)(16,43,45,35)(17,39,46,55)(21,49,48,31)(22,51,54,36)(23,42,30,37)(26,32,47,58)(27,40,28,41)(44,63,52,59)(50,60,64,57)(53,61,56,62);(1,3)(2,9)(4,21)(5,26)(6,20)(7,18)(8,19)(10,35)(11,39)(12,34)(13,15)(14,33)(16,23)(17,27)(22,52)(24,48)(25,55)(28,46)(29,47)(30,45)(31,37)(32,40)(36,60)(38,43)(41,58)(42,49)(44,54)(50,53)(51,57)(56,64)(59,61)(62,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36)(12,37)(13,42)(14,38)(15,35)(17,44)(18,21)(19,45)(20,48)(25,51)(26,50)(27,53)(28,56)(29,54)(32,57)(33,49)(34,43)(39,59)(40,61)(41,62)(46,52)(47,64)(55,63)(58,60);(1,5,6,27,8,29,7,28)(2,11,12,40,14,25,13,41)(3,17,18,26,19,46,20,47)(4,22,23,53,24,54,30,56)(9,32,15,39,33,58,34,55)(10,36,37,61,38,51,42,62)(16,44,21,50,45,52,48,64)(31,57,35,59,49,60,43,63)Polycyclic presentation:
2^9|a^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325673);