Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2023 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24)(25,28)(26,29)(27,30)(31,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,22)(23,29)(24,25)(30,31);(10,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,29)(24,30)(25,31)(28,32);(1,3,4,7)(2,6,5,8)(18,24)(21,28)(22,25)(23,30)(27,32)(29,31);(19,26)(23,29)(27,32)(30,31)Polycyclic presentation:
2^9|b^2=g,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^c=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536843);