Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,6)(5,9)(7,11)(8,13)(12,15)(14,16)(17,18,22,25,23,19,21,27)(20,26,29,32,30,24,28,31);(2,6,9,8)(7,14,10,15)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(1,3)(2,7)(4,12)(5,11)(6,14)(8,15)(9,10)(13,16)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32);(3,11)(7,10)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90152);