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 | |
5203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,12)(11,16,15,13)(17,18)(19,20)(21,22)(23,24);(1,2)(3,8)(4,5)(6,7)(10,12)(13,16)(18,22)(19,23);(2,7)(3,4)(10,12)(11,15)(17,19,21,23)(18,20,22,24);(1,3,6,4)(2,8,7,5)(9,11)(10,13)(12,16)(14,15)(17,20)(18,23)(19,22)(21,24);(1,4,6,3)(2,5,7,8)(9,12,14,10)(11,16,15,13);(1,5)(2,3)(4,7)(6,8)(9,13,14,16)(10,15,12,11)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706681);