Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,12,17,13,16,20,15)(11,14,19,23,18,21,24,22);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,21)(17,23)(20,24);(1,3,4,7)(2,6,5,8)(14,21)(22,23);(2,5)(6,8)(10,15)(12,20)(14,22)(16,17)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252551, #72252557, #72252565, … |
| maximal normal subgroup of | #72252551, #72252557, #72252565, … |
| maximal quotient of | #72302309, C2×#449748, #72316748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356944);