Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119572 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,10,12,7)(4,6,13,8)(11,15,16,14)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,32,31)(26,30);(2,6)(3,11)(4,13)(7,10)(8,9)(12,16)(17,19)(18,22)(20,25)(21,26)(23,31)(24,30)(27,29)(28,32);(1,3,4,11,5,12,13,16)(2,7,8,15,9,10,6,14)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,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, 26768);