Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14519 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1262 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)(3,15,20,33)(4,21)(5,11,29,41)(6,12)(7,13)(8,14)(9,18,34,19)(10,25)(16,31,45,59)(17,38)(22,35,48,58)(23,51,54,49)(24,37)(26,42)(27,55)(28,39,30,40)(32,46)(36,52,61,53)(43,47)(44,57)(50,60)(56,62)(63,64);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,35)(11,38)(12,33)(13,15)(14,34)(16,24)(17,28)(21,31)(23,44)(25,45)(26,48)(29,46)(30,47)(32,39)(36,57)(37,58)(40,55)(41,43)(42,59)(49,62)(50,52)(51,60)(53,56)(54,63)(61,64);(1,4,6,24,8,26,7,25)(2,10,12,21,14,37,13,42)(3,16,19,48,20,45,18,22)(5,23,30,53,29,54,28,52)(9,31,33,58,34,59,15,35)(11,36,40,49,41,61,39,51)(17,44,27,56,47,63,46,50)(32,57,38,62,55,64,43,60);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32)(10,36)(12,40)(13,39)(14,41)(15,43)(16,44)(18,46)(19,27)(20,47)(21,49)(22,50)(24,53)(25,52)(26,54)(31,57)(33,38)(34,55)(35,60)(37,61)(42,51)(45,63)(48,56)(58,62)(59,64)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295558);