Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,20)(12,17,26,36)(13,16)(14,19)(15,22,31,24)(18,29)(21,34)(23,39)(25,35,28,33)(27,37)(30,40,32,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,24)(13,27)(14,21)(16,32)(17,34)(18,30)(19,31)(20,36)(22,26)(23,29)(25,38)(28,40)(33,39)(35,37);(1,3)(2,5)(4,7)(6,8)(10,16,19,18)(11,21)(15,30,31,32)(17,35,36,33)(20,37,34,39)(22,24)(23,27)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72285825 |
| maximal normal subgroup of | #72285825 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40659);