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 | |
5096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,14)(11,12,15,16)(18,22)(23,24);(1,2,6,4)(3,8,5,7)(17,18,21,22)(19,23,20,24);(1,3,6,5)(2,7,4,8)(10,14)(11,15)(17,19,21,20)(18,23,22,24);(1,2,6,4)(3,8,5,7)(9,11)(10,12)(13,15)(14,16)(17,20,21,19)(18,24,22,23);(1,4,6,2)(3,8,5,7)(9,12)(10,11)(13,16)(14,15);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7806423);