Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #2856 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(10,13,14,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,2,5,6)(3,7,4,8)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,14)(13,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=eh,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,f^c=fi,g^c=ghi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 131772);