Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14299 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1356 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)(4,18,21,9)(5,10)(6,25)(7,12)(8,16)(11,28)(14,39,43,57)(15,40)(17,37)(19,49,50,32)(20,33,27,38)(22,34,52,61)(23,53)(24,36)(26,31)(29,42,48,47)(30,45)(35,55)(41,60,63,58)(44,64,51,56)(46,62)(54,59);(1,3)(2,8)(4,14)(5,15)(6,26)(7,17)(9,29)(10,30)(11,37)(12,31)(13,25)(16,28)(18,48)(19,41)(20,47)(21,43)(22,51)(23,54)(24,46)(27,42)(32,56)(33,39)(34,60)(35,62)(36,59)(38,57)(40,53)(44,52)(45,55)(49,64)(50,63)(58,61);(1,4)(2,9)(3,14)(5,22)(6,27)(7,21)(8,29)(10,34)(11,38)(12,18)(13,39)(15,44)(16,47)(17,43)(19,23)(20,28)(24,52)(25,33)(26,42)(30,58)(31,48)(32,35)(36,61)(37,57)(40,56)(41,45)(46,51)(49,53)(50,55)(54,63)(59,60)(62,64);(1,5)(2,10)(3,15)(4,19,21,50)(6,23)(7,24)(8,30)(9,32,18,49)(11,35)(12,36)(13,40)(14,41,43,63)(16,45)(17,46)(20,22,27,52)(25,53)(26,54)(28,55)(29,56,48,64)(31,59)(33,34,38,61)(37,62)(39,60,57,58)(42,44,47,51)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288676);