Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| 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 | |
1885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,9,13,10,8,12,17)(7,11,15,19,14,16,18,20);(1,2,4,3)(5,7)(6,11)(8,16)(9,15)(10,14)(12,18)(13,19)(17,20);(5,8,9,17,10,6,12,13)(7,11,18,20,14,16,15,19);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72252463, #72252467, D4×#1891, … |
| maximal normal subgroup of | #72252463, #72252467, D4×#1891, … |
| maximal quotient of | #72279440, #72343087 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281589);