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 | |
5913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,12)(11,15,13,16);(1,3,7,6)(2,4,8,5)(9,11,14,13)(10,15,12,16)(17,18)(19,20)(21,23)(22,24);(1,4,7,5)(2,3,8,6)(9,11,14,13)(10,16,12,15)(17,19,21,24)(18,22,23,20);(2,8)(4,5)(19,24)(20,22);(1,5,7,4)(2,3,8,6)(9,12,14,10)(11,15,13,16)(17,20)(18,19)(21,22)(23,24);(1,6,7,3)(2,5,8,4)(9,13,14,11)(10,15,12,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72256345, #72263024, #72263030 |
| maximal normal subgroup of | #72256345, #72263024, #72263030 |
| maximal quotient of | #72263313 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9046663);