Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 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)(17,18,22,19)(20,28,30,32)(21,26,23,25)(24,31,27,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,27,30,24)(21,26,23,25)(28,29,32,31);(10,14)(13,16)(18,19)(24,27)(25,26)(28,32);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(17,21,22,23)(18,25,19,26)(20,29,30,31)(24,28,27,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536833);