Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624128 |
| 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 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1536 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,23)(13,28)(14,16)(15,21)(17,26)(18,25)(20,35)(22,39)(24,34)(27,36)(29,33)(30,37)(31,40)(32,38);(1,3)(2,5)(4,8)(6,7)(11,20)(15,30)(19,35)(21,37)(22,29)(24,40)(31,34)(33,39);(9,11,14,20)(10,15,16,30)(12,24,27,40)(13,29,25,22)(17,33,32,39)(18,34,28,31)(19,36,35,23)(21,38,37,26);(3,8)(5,7)(10,16)(15,30)(17,32)(18,28)(19,35)(23,36)(31,34)(33,39);(2,6)(5,7)Polycyclic presentation:
2^9|c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531338);