Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162948 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,13,14,15)(17,18,22,26,23,20,21,25)(19,24,28,31,30,27,29,32);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(3,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,23,18)(19,27,30,24)(21,26,22,25)(28,32,29,31);(2,6)(5,7)(17,21,23,22)(18,25,20,26)(19,28,30,29)(24,31,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122753);