Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3272 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,12)(7,11,9,8)(13,14)(15,18)(16,19)(17,20);(6,8)(7,9)(11,12)(13,15,17,16)(14,18,20,19);(5,7,10,9)(6,11,12,8)(14,19)(15,16)(18,20);(5,8)(6,7)(9,12)(10,11)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=h,f^2=i,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.
| maximal subgroup of | #72279809, #72279909, #72280170 |
| maximal normal subgroup of | #72279809, #72279909, #72280170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 129632);