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 | |
5637 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)(3,8)(4,7)(5,6)(10,14)(12,15);(1,3,5,7)(2,4,6,8)(10,14)(11,16)(17,18)(19,22)(20,23)(21,24);(9,10,13,14)(11,15,16,12)(17,19,21,23)(18,20,24,22);(2,6)(4,8)(11,16)(12,15)(17,19,21,23)(18,22,24,20);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,20)(21,24)(22,23);(1,4)(2,7)(3,6)(5,8)(9,12,13,15)(10,11,14,16)(17,20)(18,23)(19,24)(21,22)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^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,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72303495, #72306585 |
| maximal normal subgroup of | #72303495, #72306585 |
| maximal quotient of | #72303520, #72305961 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9133402);