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 | |
6486 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)(11,14,12,16)(17,18)(19,20)(21,24)(22,23);(1,2,4,5)(3,6,7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,24)(21,23);(2,5)(3,7)(10,15)(11,12)(17,18)(19,22)(20,23)(21,24);(2,5)(6,8)(11,12)(14,16)(17,20,21,22)(18,23,24,19);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,15,14)(17,20)(18,19)(21,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948168);