Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1533 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,20,15,19,31,38,16,22,32,18,14,21,12,29)(11,23,25,37,27,36,24,40,28,35,30,39,26,17,33,34);(2,5)(3,8)(6,7)(9,11)(10,17)(12,30)(13,24)(14,27)(15,26)(16,28)(18,37)(19,35)(20,39)(21,23)(22,36)(25,31)(29,34)(32,33)(38,40);(9,12,14,32,16,31,15,13)(10,18,19,29,22,20,21,38)(11,24,26,25,28,33,27,30)(17,34,23,37,36,40,35,39)Polycyclic presentation:
2^9|a^2=e,c^2=fg,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32730);