Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 94 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)(4,6)(7,8)(9,10)(11,21,23,38,25,31,24,35)(12,26)(13,18)(14,19)(15,20)(16,22,32,39,34,40,33,29)(17,28)(27,37)(30,36);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,29)(14,23)(15,25)(17,31)(18,35)(19,32)(20,34)(21,26)(24,30)(27,39)(28,40)(33,36)(37,38);(2,6)(5,7)(9,12,14,27,15,28,30,13)(10,17,19,18,20,26,36,37)(11,22,23,39,25,40,24,29)(16,31,32,35,34,21,33,38)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31646);