Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14255 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2060 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)(3,13,17,8)(4,18,22,9)(5,10)(6,12)(7,11)(14,28)(15,41,44,29)(16,38,25,30)(19,31)(20,34,49,32)(21,39,26,33)(23,35)(24,36)(27,37)(40,43)(42,54)(45,62,52,56)(46,47)(48,57)(50,60,53,58)(51,61)(55,63)(59,64);(1,3)(2,8)(4,19)(5,15)(6,25)(7,17)(9,31)(10,29)(11,13)(12,38)(14,21)(16,27)(18,47)(20,48)(22,46)(23,44)(24,52)(26,43)(28,33)(30,37)(32,57)(34,59)(35,41)(36,62)(39,40)(42,50)(45,51)(49,64)(53,63)(54,58)(55,60)(56,61);(1,4)(2,9)(3,14)(5,20)(6,26)(7,22)(8,28)(10,34)(11,18)(12,39)(13,40)(15,42)(16,46)(17,43)(19,25)(21,27)(23,49)(24,53)(29,55)(30,47)(31,38)(32,35)(33,37)(36,58)(41,54)(44,63)(45,64)(48,52)(50,51)(56,57)(59,62)(60,61);(1,5)(2,10)(3,15)(4,20)(6,24)(7,23)(8,29)(9,32)(11,35)(12,36)(13,41)(14,42)(16,45)(17,44)(18,34)(19,48)(21,50)(22,49)(25,52)(26,53)(27,51)(28,54)(30,56)(31,57)(33,58)(37,61)(38,62)(39,60)(40,55)(43,63)(46,64)(47,59)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287027);