Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482357 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,21,19,17)(13,22,23,15)(14,18)(20,24);(2,5)(3,8)(6,7)(10,15)(16,21)(17,24)(18,22);(2,6)(5,7)(9,11,14,20)(10,16,18,24)(12,13,19,23)(15,21,22,17);(2,6)(5,7)(9,12,14,19)(10,17,18,21)(11,13,20,23)(15,24,22,16)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72281069, #72281110, #72281283, … |
| maximal normal subgroup of | #72281069, #72281110, #72281283, … |
| maximal quotient of | #72283051, #72293137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389557);