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 | |
5912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22);(9,11)(10,14)(12,16)(13,15)(18,22)(19,24);(1,2,5,4)(3,7,8,6)(9,12)(10,15)(11,16)(13,14)(17,19)(18,20)(21,24)(22,23);(17,20)(18,19)(21,23)(22,24);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,15)(14,16);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,16)(13,14)Polycyclic presentation:
2^9|c^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572037);