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×#3274 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1270 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)(6,10)(7,8)(11,12)(13,14)(15,18)(16,19)(17,20);(5,6,9,11)(7,10,8,12)(13,15,17,16)(14,18,20,19);(6,11)(10,12)(14,19)(15,16)(18,20);(5,7)(8,9)(10,12)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dhi,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 | #72279759, #72279872, #72280170 |
| maximal normal subgroup of | #72279759, #72279872, #72280170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128189);