Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124940 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,13,20,16,22,30,33,14,23,29,17,15,21,12,19)(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);(9,11)(10,18)(12,24)(13,31)(14,26)(15,27)(16,28)(17,32)(19,34)(20,38)(21,35)(22,36)(23,37)(25,29)(30,39)(33,40)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240720, #72240794, #72240795, … |
| maximal normal subgroup of | #72240720, #72240794, #72240795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32136);