Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #2424 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1669 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16);(3,8)(5,7)(10,14)(13,16);(2,6)(5,7)(19,26)(22,25)(27,29)(30,32);(17,19,20,27,21,26,28,29)(18,22,23,30,24,25,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285498);