Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #388 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5)(4,6,8,7)(10,15)(11,16)(12,13)(17,18,20,23,21,19,22,24);(2,5)(6,7)(9,10,13,11,14,16,12,15)(17,19,20,24,21,18,22,23);(2,6)(4,8)(5,7)(9,11,13,16,14,15,12,10)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=dh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^c=egh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216350, #72216483, #72216495 |
| maximal normal subgroup of | #72216350, #72216483, #72216495 |
| maximal quotient of | #72216338, #72216476, #72225479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9102);