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 | #2541 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1426 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)(17,18,21,27,22,25,31,26)(19,28,30,24,20,23,29,32);(1,2,6,3)(4,7,5,8)(9,10,13,14)(11,16,12,15)(17,19,22,20)(18,23,25,28)(21,30,31,29)(24,27,32,26);(1,3,6,2)(4,7,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,19)(18,24,25,32)(21,30,31,29)(23,27,28,26);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,15,14,16)(18,25)(19,20)(26,27)(29,30);(1,5,6,4)(2,7,3,8)(17,21,22,31)(18,26,25,27)(19,29,20,30)(23,32,28,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,d^2=fi,e^2=fh,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, 6876983);