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 | |
5885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,13)(11,16,12,15)(17,18)(19,23)(20,21)(22,24);(1,2)(3,7)(4,5)(6,8)(9,11,14,12)(10,15,13,16);(2,5)(6,7)(11,12)(15,16)(18,21)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,13,15)(17,18)(19,22)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,19)(18,22)(20,24)(21,23);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,15,12,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7640340);