Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1556 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)(9,10,13,15)(11,14,12,16)(18,22)(19,26)(24,25)(28,30);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18,20,22)(19,27,26,32)(21,24,29,25)(23,30,31,28);(9,12,13,11)(10,14,15,16)(17,19,20,26)(18,23,22,31)(21,30,29,28)(24,27,25,32);(18,24)(21,29)(22,25)(23,27)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(19,26)(23,31)(27,32)(28,30)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355649);