Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(1,2,4,6)(3,7,8,5)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24);(10,14)(12,21)(15,22)(16,17)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252480, #72252481, #72252487, … |
| maximal normal subgroup of | #72252480, #72252481, #72252487, … |
| maximal quotient of | C2×#362781, #72314076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269977);