Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124851 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 403 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)(4,6)(7,8)(9,10,13,17,16,21,12,19,14,22,31,34,15,20,30,33)(11,23,25,38,28,37,29,40,26,36,39,35,27,18,24,32);(2,5)(3,8)(6,7)(10,17)(12,13)(15,16)(18,32)(19,20)(21,33)(22,34)(23,35)(24,39)(25,29)(27,28)(30,31)(36,38)(37,40);(2,6)(5,7)(9,11,14,26)(10,18)(12,29,30,24)(13,25,31,39)(15,27,16,28)(17,32)(19,38)(20,36)(21,23)(22,37)(33,35)(34,40)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240793, #72240794, #72240797, … |
| maximal normal subgroup of | #72240793, #72240794, #72240797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32109);