Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25,27,39)(13,29)(14,17)(15,19)(16,22)(18,28,32,26)(21,35,37,31)(23,33)(24,30)(34,38,40,36);(1,3,4,7)(2,5,6,8)(9,11,15,24,14,23,13,22)(10,16,29,33,17,30,19,20)(12,21,28,38,27,37,26,36)(18,31,39,40,32,35,25,34);(2,6)(5,8)(10,17)(16,30)(18,32)(19,29)(20,33)(25,39)(31,35)(34,40);(9,12,13,26,14,27,15,28)(10,18,19,25,17,32,29,39)(11,21,22,36,23,37,24,38)(16,31,20,34,30,35,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240719, #72240721, #72240722, … |
| maximal normal subgroup of | #72240719, #72240721, #72240722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299603);