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 | #42589 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 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)(23,29)(24,26)(31,32);(10,13)(11,16)(18,23)(19,25)(21,28)(22,29)(27,30)(31,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,18)(19,26)(20,22)(21,29)(23,28)(24,27)(25,32)(30,31);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(19,27)(24,26)(25,30)(31,32);(17,19,20,27)(18,24,22,26)(21,30,28,25)(23,31,29,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6608707);