Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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 | #25397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 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)(10,14)(11,12)(17,18,21,26,22,24,31,25)(19,20,28,30,27,23,32,29);(1,2,6,3)(4,7,5,8)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,24)(21,28)(22,27)(25,30)(26,29)(31,32);(1,3,6,2)(4,7,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,29)(22,23)(24,27)(25,28)(26,32)(30,31);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,15,14,16)(18,24)(19,27)(25,26)(28,32);(1,5,6,4)(2,7,3,8)(17,21,22,31)(18,25,24,26)(19,28,27,32)(20,29,23,30)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6876996);