Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14345 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1139 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)(3,8)(4,17,20,44)(5,10)(6,11)(7,12)(9,21,31,48)(13,36,18,45)(14,27)(15,28)(16,29)(19,43,22,49)(23,33)(24,34)(25,35)(26,38,53,46)(30,47,32,50)(37,59,39,61)(40,55)(41,56)(42,57)(51,58)(52,60,54,62)(63,64);(1,3)(2,8)(4,18)(5,14)(6,15)(7,16)(9,26)(10,27)(11,28)(12,29)(13,20)(17,36)(19,39)(21,46)(22,37)(23,40)(24,41)(25,42)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(38,48)(43,59)(44,45)(47,62)(49,61)(50,60)(51,63)(58,64);(1,4)(2,9,12,31)(3,13,16,18)(5,22,24,19)(6,21)(7,20)(8,26)(10,32)(11,17,35,44)(14,39)(15,38,42,46)(23,50,51,47)(25,48)(27,54,56,52)(28,36)(29,53)(30,34)(33,49)(37,41)(40,62)(43,58)(45,57)(55,61,64,59)(60,63);(1,5,6,23,7,24,25,51)(2,10,11,33,12,34,35,58)(3,14,15,40,16,41,42,63)(4,19,21,47,20,22,48,50)(8,27,28,55,29,56,57,64)(9,30,17,43,31,32,44,49)(13,37,38,60,18,39,46,62)(26,52,36,59,53,54,45,61)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=f,f^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285536, #72285582, #72286021, … |
| maximal normal subgroup of | #72285536, #72285582, #72286021, … |
| maximal quotient of | C2×#382864 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290060);