Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1611 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,28,24,29,23,41)(19,30,34,44,35,43,33,25)(20,36,37,26,32,45,38,46)(21,27,39,42,40,48,31,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,31)(21,38)(22,33)(23,34)(24,35)(26,42)(27,45)(28,43)(29,44)(30,41)(32,39)(36,48)(37,40)(46,47);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20,24,32)(18,26,29,46)(19,31,35,39)(21,33,40,34)(22,37,23,38)(25,42,44,47)(27,30,48,43)(28,45,41,36);(4,12)(7,16)(9,11)(13,15)(17,21,22,39,24,40,23,31)(18,27,28,42,29,48,41,47)(19,32,34,38,35,20,33,37)(25,26,30,45,44,46,43,36);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,29,41)(19,33,35,34)(20,37,32,38)(21,31,40,39)(25,43,44,30)(26,45,46,36)(27,47,48,42)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7012004);