Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1540 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(18,22)(19,26)(23,25)(27,32);(10,13)(11,16)(17,18)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(18,25)(19,26)(21,28)(22,23)(24,30)(29,31)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6608662);