Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,29)(21,26)(22,31)(23,27)(25,30)(28,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(11,15)(13,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(9,11)(10,13)(12,15)(14,16)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25);(11,15)(13,16)(18,27)(24,29)(25,32)(26,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6499290);