Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62793 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4,11,7)(5,13)(6,9)(8,15)(10,14,16,12)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,13)(15,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,4,6,7)(2,3,9,11)(5,14,15,12)(8,16,13,10);(1,5,6,15)(2,8,9,13)(3,10,11,16)(4,12,7,14)(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,e^c=eh,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93070);