Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30471 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,21)(22,29)(23,28)(24,30)(26,32)(27,31);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(18,20)(19,26)(24,29)(28,31);(11,16)(13,15)(17,19)(18,23)(20,27)(21,26)(22,31)(24,32)(25,29)(28,30);(2,3)(6,7)(10,14)(13,15)(17,20,21,18)(19,27,26,23)(22,29,30,24)(25,28,32,31);(1,4,5,8)(2,6,3,7)(18,20)(23,27)(24,29)(25,32)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6560429);