Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #372790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14100 |
| 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 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,26)(3,9,17,32,16,33,28,31)(4,18,22,10,21,36,27,35)(5,11,25,37,23,38,24,39)(14,40,43,56,19,49,42,29)(15,44,45,57,46,30,47,58)(20,51,52,34,53,48,54,59)(41,60,62,55,50,64,61,63);(1,3,6,17,8,16,7,28)(2,9,13,31,12,33,26,32)(4,19,22,43,21,14,27,42)(5,15,25,47,23,46,24,45)(10,29,18,40,35,56,36,49)(11,30,39,58,38,44,37,57)(20,50,54,61,53,41,52,62)(34,55,48,60,59,63,51,64);(1,4)(2,10)(3,14)(5,20)(6,27)(7,22)(8,21)(9,29)(11,34)(12,35)(13,36)(15,41)(16,19)(17,43)(18,26)(23,53)(24,54)(25,52)(28,42)(30,55)(31,49)(32,40)(33,56)(37,48)(38,59)(39,51)(44,63)(45,62)(46,50)(47,61)(57,60)(58,64);(1,5,6,25,8,23,7,24)(2,11,13,39,12,38,26,37)(3,15,17,47,16,46,28,45)(4,20,27,52,21,53,22,54)(9,30,31,58,33,44,32,57)(10,34,36,51,35,59,18,48)(14,41,43,61,19,50,42,62)(29,55,49,64,56,63,40,60)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=efg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279989);