Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | |
1680 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)(19,25)(20,32)(21,27)(22,38)(23,28)(24,29)(26,34)(30,43)(31,44)(33,41)(35,46)(36,48)(37,45)(39,42)(40,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,25)(20,33)(21,35)(22,30)(23,39)(24,31)(26,45)(27,46)(28,42)(29,44)(32,41)(34,37)(36,40)(38,43)(47,48);(17,20,24,36)(18,26,29,46)(19,21,31,37)(22,35,23,34)(25,41,44,47)(27,42,45,43)(28,48,38,32)(30,40,39,33);(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)(17,21,24,37)(18,27,29,45)(19,20,31,36)(22,33,23,40)(25,32,44,48)(26,42,46,43)(28,47,38,41)(30,34,39,35)Polycyclic presentation:
2^9|c^2=i,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582725);