Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 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)(4,5)(7,8)(17,18,21,26)(19,25,22,23)(20,28,30,32)(24,27,29,31);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,21,22)(18,23,26,25)(20,27,30,31)(24,32,29,28);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,19,27,21,30,22,31)(18,24,23,32,26,29,25,28);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,18)(19,23)(20,29)(21,26)(22,25)(24,30)(27,28)(31,32);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,19,21,22)(18,25,26,23)(20,27,30,31)(24,28,29,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822967);