Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14513 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1017 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)(3,14,18,9)(4,19,22,38)(5,24)(6,13)(7,29)(8,12)(10,23,39,28)(11,31)(15,44)(16,48,25,56)(17,35,30,34)(20,42)(21,37,55,62)(26,40)(27,41)(32,47)(33,50,43,49)(36,46)(45,63)(51,54,57,53)(52,60)(58,59)(61,64);(1,3)(2,9)(4,20)(5,25)(6,17)(7,30)(8,18)(10,36)(11,33)(12,14)(13,34)(15,23)(16,26)(19,42)(21,52)(22,47)(24,56)(27,50)(28,46)(29,35)(31,49)(32,38)(37,61)(39,44)(40,48)(41,43)(45,53)(51,59)(54,58)(55,64)(57,63)(60,62);(1,4)(2,10)(3,15,18,46)(5,21)(6,28)(7,23)(8,22)(9,32,14,42)(11,37)(12,39)(13,38)(16,45,25,58)(17,20,30,47)(19,29)(24,57)(26,55)(27,54)(31,53)(33,59,43,63)(34,44,35,36)(40,51)(41,62)(48,60,56,61)(49,52,50,64);(1,5)(2,11)(3,16,18,25)(4,21)(6,27)(7,31)(8,26)(9,33,14,43)(10,37)(12,41)(13,24)(15,45,46,58)(17,49,30,50)(19,51)(20,52,47,64)(22,55)(23,53)(28,54)(29,40)(32,59,42,63)(34,56,35,48)(36,60,44,61)(38,57)(39,62)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295279);