Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| prime radical | #33520 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,23)(13,26)(14,18)(15,20)(17,24)(19,28)(21,37)(22,36)(25,39)(27,31)(29,34)(30,35)(32,40)(33,38);(2,5)(6,8)(9,11,15,14)(10,16,20,18)(12,21,25,30)(13,22,29,28)(17,31,33,32)(19,26,36,34)(23,37,39,35)(24,27,38,40);(9,12)(10,17)(11,21)(13,27)(14,30)(15,25)(16,31)(18,32)(19,35)(20,33)(22,38)(23,26)(24,28)(29,40)(34,39)(36,37);(1,3,4,7)(2,6,5,8);(9,11,15,14)(10,18,20,16)(12,21,25,30)(13,22,29,28)(17,32,33,31)(19,34,36,26)(23,35,39,37)(24,27,38,40)Polycyclic presentation:
2^9|b^2=i,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287749, #72287755, #72287767, … |
| maximal normal subgroup of | #72287749, #72287755, #72287767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530297);