Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1098 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)(9,10,14,17,15,19,11,20)(12,16,21,24,13,18,22,23);(2,5)(3,8)(6,7)(10,16)(12,13)(17,23)(18,19)(20,24)(21,22);(2,6)(5,7)(9,11,15,14)(10,17,19,20)(12,21,13,22)(16,23,18,24);(2,6)(5,7)(9,12,15,13)(10,16,19,18)(11,21,14,22)(17,23,20,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280943, #72281058, #72281460, … |
| maximal normal subgroup of | #72280943, #72281058, #72281460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389587);