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 | |
6020 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)(3,4)(5,8)(6,7)(17,18)(19,20)(21,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,19,21,23)(18,20,24,22);(1,4)(2,7)(3,8)(5,6)(9,11,13,16)(10,14,15,12)(17,19)(18,22)(20,24)(21,23);(2,8)(4,6)(11,16)(12,14)(17,19,21,23)(18,20,24,22);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,23)(19,24)(21,22)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9044691);