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 | |
6079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(4,8)(9,10,14,12)(11,15,13,16)(18,21)(19,23);(2,3)(4,8)(9,11,14,13)(10,15,12,16)(17,18,22,21)(19,24,23,20);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,12,15)(17,19)(18,20)(21,24)(22,23);(2,3)(6,7)(17,18,22,21)(19,20,23,24);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,20,22,24)(18,23,21,19);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72260809 |
| maximal normal subgroup of | #72260809 |
| maximal quotient of | #72260830 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736063);