Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
892 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 892 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:
(1,2)(3,7,8,5)(4,6)(9,10,14,19,15,18,13,28)(11,20,23,34,24,16,22,31)(12,25,27,17,26,33,29,32)(21,36,38,35,37,40,39,30);(1,3)(2,5)(4,8)(6,7)(9,11,15,24)(10,16,18,20)(12,21,26,37)(13,22,14,23)(17,30,32,35)(19,34,28,31)(25,36,33,40)(27,38,29,39);(9,12)(10,17)(11,21)(13,29)(14,27)(15,26)(16,30)(18,32)(19,33)(20,35)(22,39)(23,38)(24,37)(25,28)(31,36)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284952, #72285112, #72285486, … |
| maximal normal subgroup of | #72284952, #72285112, #72285486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40553);