Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,3)(4,8,5,7)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3,6,2)(4,8,5,7)(9,10,12,14)(11,15,16,13)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(1,4,6,5)(2,7,3,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,26)(27,29)(28,30)(31,32);(1,4,6,5)(2,8,3,7)(11,16)(13,15)(18,23)(22,25)(24,30)(29,31);(1,5,6,4)(2,8,3,7)(10,14)(13,15)(19,26)(22,25)(27,32)(29,31);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008366);