Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,25,21,26,33,35,19,27,31,36,32,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,27)(17,25)(18,26)(22,33)(23,31)(24,32)(28,36)(29,37)(30,35)(34,39)(38,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(10,16)(11,19)(17,28)(18,29)(20,31)(21,32)(30,38)(33,39)Polycyclic presentation:
2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72231978 |
| maximal normal subgroup of | #72231978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529916);