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 |
| prime radical | #61423 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,29,24,28,23,40)(19,30,33,45,32,44,38,25)(20,34,36,26,37,47,41,43)(21,27,31,46,39,48,35,42);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,32)(18,25,28,45)(20,35,37,31)(21,36,39,41)(22,38,23,33)(26,46,43,42)(27,34,48,47)(29,44,40,30);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,31)(21,38)(22,36)(23,41)(24,37)(25,42)(27,30)(28,43)(29,47)(32,35)(33,39)(34,40)(44,48)(45,46);(3,10)(6,9)(11,16)(14,15)(17,21,22,31,24,39,23,35)(18,27,29,46,28,48,40,42)(19,20,33,36,32,37,38,41)(25,43,30,34,45,26,44,47);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,28)(19,32)(21,39)(26,43)(29,40)(31,35)(33,38)(34,47)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=f,e^2=hi,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, 7088498);