Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1810 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,22)(19,25)(20,24)(26,32)(28,29)(30,31);(2,5)(6,8)(17,18,21,22)(19,23,25,27)(20,24,28,29)(26,30,31,32);(9,11,12,16)(10,13,14,15)(17,19,21,25)(18,23,22,27)(20,26,28,31)(24,30,29,32);(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(1,3,4,7)(2,6,5,8)(19,25)(23,27)(26,31)(30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,e^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, 6537625);