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 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,23)(19,21)(20,27)(24,31)(25,32)(26,29)(28,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,24)(23,31)(25,30)(28,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,18)(19,25)(20,28)(21,30)(22,24)(23,27)(26,31)(29,32);(3,11)(6,9)(10,16)(13,15)(17,19,22,21)(18,25,24,30)(20,26,29,27)(23,28,31,32);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^9|c^2=f,d^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, 7531475);