Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667700 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1673 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)(3,7)(4,6)(5,8)(17,18,22,25)(19,26,27,32)(20,24,21,29)(23,30,31,28);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,13)(14,16)(19,27)(23,31)(26,32)(28,30);(3,8)(5,7)(10,13)(14,15)(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,31)(26,29);(2,6)(5,7)(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699781);