Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,15,11)(12,22,25,37)(13,26,30,39)(14,21,28,18)(16,32,19,23)(17,34,20,29)(24,38,31,35)(27,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,11,15,10)(12,19,25,16)(13,20,30,17)(14,21,28,18)(22,23,37,32)(24,36,31,33)(26,29,39,34)(27,38,40,35);(9,12,15,25)(10,16,11,19)(13,27,30,40)(14,31,28,24)(17,35,20,38)(18,36,21,33)(22,39,37,26)(23,34,32,29);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,11)(16,19)(17,20)(18,21)(22,37)(26,39)(33,36)(35,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72322867 |
| maximal normal subgroup of | #72322867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531305);