Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | #60694 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1564 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)(17,18,22,26)(19,25,31,24)(20,33,32,41)(21,35,38,27)(23,29,37,28)(30,40,46,42)(34,47,36,43)(39,44,48,45);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19,22,31)(18,24,26,25)(20,23,32,37)(21,36,38,34)(27,43,35,47)(28,41,29,33)(30,39,46,48)(40,45,42,44);(17,20,23,31,22,32,37,19)(18,25,29,33,26,24,28,41)(21,34,39,46,38,36,48,30)(27,42,45,43,35,40,44,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,26)(24,25)(27,35)(28,29)(33,41)(40,42)(43,47)(44,45);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,37,48,22,38,23,39)(18,27,28,44,26,35,29,45)(19,30,32,36,31,46,20,34)(24,40,33,43,25,42,41,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fi,e^2=fh,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719070);