Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| 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 | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,13,10,7,11,15)(8,14,17,20,16,12,18,19);(5,7,9,15,10,6,11,13)(8,14,18,19,16,12,17,20);(1,2,3,4);(5,8)(6,12)(7,14)(9,18)(10,16)(11,17)(13,20)(15,19)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^b=gi,e^d=ei,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72252465, #72252468, D4×#1891, … |
| maximal normal subgroup of | #72252465, #72252468, D4×#1891, … |
| maximal quotient of | #72257969, #72264440, #72279440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300254);