Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133422 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2309 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 680 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)(9,10,13,18,14,17,26,32)(11,19,21,33,22,15,27,29)(12,23,25,37,24,16,30,31)(20,28,36,39,35,34,38,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,17,19)(12,20,24,35)(13,21,26,27)(16,28,23,34)(18,33,32,29)(25,36,30,38)(31,40,37,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,24)(10,16)(11,20,22,35)(13,25,26,30)(15,28)(17,23)(18,31)(19,34)(21,36,27,38)(29,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=eh,f^b=fg,d^c=dh,e^c=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284785, #72284788, #72284804, … |
| maximal normal subgroup of | #72284785, #72284788, #72284804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40626);