Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495737 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21452 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15)(4,21,24,41)(5,25,27,40)(6,14,8,12)(9,31)(10,30,37,23)(11,28,39,29)(16,45)(17,43)(18,35)(19,38)(20,44)(22,53,55,36)(26,33)(32,59)(34,50)(42,46)(47,60)(48,64)(49,63)(51,58)(52,57,62,56)(54,61);(1,3)(2,9)(4,16)(5,26)(6,19)(7,20)(8,17)(10,32)(11,38)(12,34)(13,35)(14,33)(15,40)(18,28)(21,51)(22,54)(23,46)(24,49)(25,44)(27,50)(29,31)(30,48)(36,45)(37,61)(39,43)(41,60)(42,62)(47,57)(52,64)(53,63)(55,59)(56,58);(1,4)(2,10)(3,16)(5,22)(6,30)(7,24)(8,23)(9,32)(11,36)(12,41)(13,37)(14,21)(15,42)(17,46)(18,47)(19,48)(20,49)(25,52)(26,54)(27,55)(28,57)(29,56)(31,58)(33,51)(34,60)(35,61)(38,45)(39,53)(40,62)(43,63)(44,64)(50,59);(3,17,20,19)(5,27)(9,33,35,34)(11,39)(15,43,44,38)(16,46,49,48)(18,50,31,26)(22,55)(25,40)(28,29)(32,51,61,60)(36,53)(42,63,64,45)(47,59,58,54)(52,62)(56,57)Polycyclic presentation:
2^9|a^2=g,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285481, #72285570, #72285805, … |
| maximal normal subgroup of | #72285481, #72285570, #72285805, … |
| maximal quotient of | #72287298, #72287501, #72287690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402934);