Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #46430 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#536 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 846 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(5,8)(9,10,14,19,15,20,13,18)(11,21,25,17,22,33,24,34)(12,26,27,35,28,16,29,31)(23,36,39,30,38,37,40,32);(1,2)(3,5)(4,6)(7,8)(10,16)(11,22)(17,32)(18,31)(19,35)(20,26)(21,36)(23,38)(24,25)(30,34)(33,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,22)(10,17,20,34)(12,23,28,38)(13,25,14,24)(16,30,26,32)(18,33,19,21)(27,40,29,39)(31,36,35,37)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,e^b=eh,f^b=fg,d^c=dh,e^c=ei,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72286157 |
| maximal normal subgroup of | #72286157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40583);