Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| 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 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,22,27)(19,30,33,25)(20,35,36,46)(21,28,37,47)(23,39,24,29)(26,40,42,32)(31,48,38,45)(34,43,41,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,21)(22,33)(23,34)(24,41)(26,45)(27,30)(28,46)(29,43)(31,32)(35,47)(36,37)(38,40)(39,44)(42,48);(17,20,24,32,22,36,23,40)(18,26,29,46,27,42,39,35)(19,31,34,21,33,38,41,37)(25,28,44,45,30,47,43,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,27)(25,30)(26,42)(28,47)(29,39)(35,46)(43,44)(45,48);(3,11)(6,9)(10,16)(14,15)(17,21,23,31,22,37,24,38)(18,28,39,48,27,47,29,45)(19,32,41,20,33,40,34,36)(25,42,43,46,30,26,44,35)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=fi,e^2=hi,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=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715964);