Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18,23,29)(19,24,34,44)(20,35,31,45)(21,38,30,26)(22,28,41,27)(25,39,46,37)(32,48,33,43)(36,47,40,42);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,36)(21,39)(22,32)(23,34)(25,26)(27,43)(28,48)(29,44)(30,37)(31,40)(33,41)(35,47)(38,46)(42,45);(17,20,23,31)(18,25,29,46)(19,30,34,21)(22,39,41,37)(24,42,44,47)(26,43,38,48)(27,45,28,35)(32,36,33,40);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,23,30)(18,26,29,38)(19,31,34,20)(22,36,41,40)(24,35,44,45)(25,43,46,48)(27,47,28,42)(32,39,33,37)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582739);