Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907961 |
| 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 | #2428 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1480 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(19,26)(22,31)(25,32)(27,30);(1,5,6,4)(2,7,3,8)(19,26)(22,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897360);