Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | D4×#121 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18,20,23)(19,21,24,22);(9,12,13,11)(10,14,15,16)(18,21)(19,24)(22,23);(2,6)(5,7);(3,8)(5,7)(18,21)(19,24)(22,23)Polycyclic presentation:
2^9|b^2=fgi,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72315352, #72315388, #72315410 |
| maximal normal subgroup of | #72315352, #72315388, #72315410 |
| maximal quotient of | #72315845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597634);