Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122155 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10)(4,6)(5,9)(7,12)(8,14)(11,15)(13,16)(17,18)(19,22,26,29)(20,23,27,31)(21,24)(25,28)(30,32);(2,6,9,8)(7,16,10,15)(17,19)(18,22,24,29)(20,25)(21,26)(23,30,31,28)(27,32);(1,3)(2,7,9,10)(4,13)(5,12)(6,15,8,16)(11,14)(17,20)(18,23,24,31)(19,25)(21,27)(22,28,29,30)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,g^b=gi,d^c=dh,e^c=eh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29351);