Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133406 |
| 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, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
708 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 708 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)(11,19)(12,23)(13,26,27,17)(14,18)(15,21,29,36)(16,25)(20,32)(22,33)(24,30,38,31)(28,39,37,34)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,18,29)(12,20,25,35)(13,21,27,36)(16,28,23,37)(17,19,26,33)(24,34,38,39)(30,40,31,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,28,29,37)(17,31,26,30)(19,32,33,40)(21,34)(22,35)(27,38)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,b^a=bd,c^a=ce,d^a=dg,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 | #72341136, #72341142, #72341150, … |
| maximal normal subgroup of | #72341136, #72341142, #72341150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40708);