Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907889 |
| 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 | #24758 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1496 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)(17,18,21,23)(19,27,28,32)(20,26,22,25)(24,30,31,29);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(18,23)(19,28)(25,26)(29,30);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,20,30,21,28,22,29)(18,24,25,32,23,31,26,27);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,27,31,32);(1,5,6,4)(2,7,3,8)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gi,d^2=fhi,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=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897320);