Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,15)(11,12,14,16)(17,18,20,22)(19,23,24,21);(1,2)(3,6)(4,7)(5,8)(9,11,13,14)(10,12,15,16)(17,19,20,24)(18,21,22,23);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,24)(22,23);(3,8)(5,6)(11,14)(12,16)(17,19)(18,21)(20,24)(22,23);(1,2)(3,5)(4,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233534, #72233536, #72233561, … |
| maximal normal subgroup of | #72233534, #72233536, #72233561, … |
| maximal quotient of | #72256742, #72256758, #72262004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8609138);