Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | #3150 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 364 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:
(2,6,7,8)(9,10,14,21,16,22,12,32,17,24,33,20,15,23,13,19)(11,25,28,18,29,36,34,38,31,35,27,39,30,40,26,37);(1,2)(3,6)(4,8)(5,7)(9,11)(10,18)(12,26)(13,34)(14,27)(15,30)(16,29)(17,31)(19,35)(20,36)(21,40)(22,38)(23,37)(24,39)(25,32)(28,33);(1,3,5,4)(2,6,7,8)(9,12,16,33,17,13,15,14)(10,19,22,21,24,32,23,20)(11,26,29,28,31,34,30,27)(18,35,38,40,39,25,37,36)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32112);