Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14520 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1142 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)(4,20,24,36)(5,25,29,11)(6,30)(7,12)(8,13)(9,17)(10,23,37,31)(15,44,21,32)(16,38,48,42)(18,34)(19,43)(22,50)(26,33,47,59)(27,39,28,40)(35,53)(41,46,58,49)(45,63)(51,57)(52,62)(54,61)(55,56)(60,64);(1,3,6,17,8,19,7,18)(2,9,12,14,13,34,30,43)(4,21,23,49,24,15,31,46)(5,26,27,48,29,47,28,16)(10,32,36,58,37,44,20,41)(11,38,39,59,25,42,40,33)(22,52,53,45,54,64,55,51)(35,60,50,63,56,62,61,57);(1,4,6,31,8,24,7,23)(2,10,12,20,13,37,30,36)(3,15,17,49,19,21,18,46)(5,22,28,53,29,54,27,55)(9,32,14,41,34,44,43,58)(11,35,40,50,25,56,39,61)(16,45,47,64,48,51,26,52)(33,57,42,60,59,63,38,62);(1,5)(2,11)(3,16)(4,22)(6,28)(7,27)(8,29)(9,33)(10,35)(12,40)(13,25)(14,42)(15,45)(17,47)(18,26)(19,48)(20,50)(21,51)(23,55)(24,54)(30,39)(31,53)(32,57)(34,59)(36,61)(37,56)(38,43)(41,60)(44,63)(46,52)(49,64)(58,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,f^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,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295652);