Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 100 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)(9,10,13,20,16,22,30,33,14,23,29,17,15,21,12,19)(11,18,25,32,28,36,24,34,26,37,31,38,27,35,39,40);(2,5)(3,8)(6,7)(10,17)(12,29)(13,30)(15,16)(18,32)(19,21)(20,23)(22,33)(24,25)(27,28)(31,39)(34,35)(36,40)(37,38);(2,6)(5,7)(9,11,14,26)(10,18,23,37)(12,24,30,39)(13,31,29,25)(15,27,16,28)(17,32,20,38)(19,34,33,40)(21,35,22,36)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240711, #72240796, #72240797, … |
| maximal normal subgroup of | #72240711, #72240796, #72240797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31461);