Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632036 |
| 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 | #24384 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1795 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(18,24)(22,26)(23,25)(30,31);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6540279);