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 | |
5246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,8)(3,5,4,7)(9,10,13,15)(11,16,14,12)(17,18,21,19)(20,23,24,22);(1,3,6,4)(2,7,8,5)(9,11)(10,12)(13,14)(15,16)(18,19)(22,23);(1,3,6,4)(2,7,8,5)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,23,24,22);(2,8)(3,4)(9,12,13,16)(10,14,15,11)(17,20,21,24)(18,22,19,23);(1,4,6,3)(2,5,8,7)(9,12)(10,11)(13,16)(14,15);(1,5)(2,3)(4,8)(6,7)(9,11)(10,12)(13,14)(15,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064572);