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 | |
5832 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,20,22,23);(2,5)(6,8)(10,12)(11,15)(17,20,21,23)(18,19,22,24);(3,7)(6,8)(11,15)(13,16)(17,18,21,22)(19,23,24,20);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,15,16)(17,20)(18,19)(21,23)(22,24);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72256737 |
| maximal normal subgroup of | #72256737 |
| maximal quotient of | #72256754 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8499377);