Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125524 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1551 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,20,27,22,24,34,39,23,28,37,44,21,26,36,43)(19,29,32,46,30,25,40,42,31,38,35,48,33,41,47,45);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(18,24)(19,30)(20,34)(21,22)(25,41)(26,28)(27,44)(31,33)(32,35)(36,37)(42,46)(45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,19,21,33,23,31,22,30)(18,25,24,38,28,41,26,29)(20,35,36,40,37,32,34,47)(27,45,39,46,44,42,43,48)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32720);