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 |
| prime radical | #31051 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18)(19,24)(20,32)(21,26)(22,27)(23,29)(25,35)(28,37)(30,41)(31,33)(34,46)(36,42)(38,43)(39,44)(40,45)(47,48);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19)(18,24)(20,29)(21,25)(22,38)(23,32)(26,35)(27,43)(28,33)(30,39)(31,37)(34,42)(36,46)(40,47)(41,44)(45,48);(17,20,23,24)(18,25,29,31)(19,26,32,28)(21,33,37,35)(22,34,40,41)(27,38,45,47)(30,42,46,44)(36,48,39,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,37)(18,26,29,28)(19,25,32,31)(20,33,24,35)(22,36,40,39)(27,42,45,44)(30,38,46,47)(34,48,41,43);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,23,40)(18,27,29,45)(19,30,32,46)(20,34,24,41)(21,36,37,39)(25,38,31,47)(26,42,28,44)(33,48,35,43)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fi,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, 6584675);