Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #61450 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,6,3,10,7)(5,8,12,16,15,11,14,13)(17,18)(19,27)(20,24)(21,28)(22,26)(23,30)(25,32)(29,31);(1,3)(2,6)(4,7)(5,13)(8,12)(9,10)(11,14)(15,16)(17,19)(18,20)(21,29)(22,23)(24,32)(25,30)(26,31)(27,28);(1,4,6,10)(2,7,3,9)(5,12,15,14)(8,13,11,16)(17,20,22,31)(18,23,26,19)(21,27,32,30)(24,25,29,28);(1,5,6,15)(2,8,3,11)(4,12,10,14)(7,13,9,16)(17,20,22,31)(18,24,26,29)(19,28,23,25)(21,30,32,27)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 129863);