Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,3)(2,5)(4,8)(6,7)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(10,14)(15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(3,8)(5,7)(9,11,13,12)(10,15,14,16);(2,6)(5,7)(9,12,13,11)(10,16,14,15)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6299064);