Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1891 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)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(10,14)(13,16)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,3,4,7)(2,6,5,8)(18,23)(22,25)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349326);