Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25731 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 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)(11,15)(12,14)(13,16)(17,18,21,24)(19,27,30,32)(20,26,22,25)(23,31,28,29);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,18)(19,28)(20,25)(21,24)(22,26)(23,30)(27,31)(29,32);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(17,19)(18,23)(20,29)(21,30)(22,31)(24,28)(25,27)(26,32);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(18,24)(23,28)(25,26)(27,32);(1,6,7,5)(2,3,8,4)(17,20,21,22)(18,25,24,26)(19,29,30,31)(23,27,28,32)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fh,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6900120);