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 | |
1539 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)(17,18,23,27)(19,20,30,24)(21,31,29,25)(22,26,32,28);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,27)(21,22)(23,30)(25,26)(28,31)(29,32);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,25,23,24,32,31)(18,21,28,30,27,29,26,19);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,21,23,29)(18,25,27,31)(19,22,30,32)(20,28,24,26);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,22,23,32)(18,26,27,28)(19,29,30,21)(20,25,24,31)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fgi,d^2=f,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, 6822977);