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 | |
5444 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)(11,16)(12,14)(17,18)(19,21)(20,23)(22,24);(1,3,5,7)(2,6,8,4)(9,10)(11,12)(13,15)(14,16)(17,18,22,24)(19,21,23,20);(1,2,5,8)(3,6,7,4)(9,11)(10,14)(12,15)(13,16)(17,19,22,23)(18,20,24,21);(1,2,5,8)(3,4,7,6)(9,10,13,15)(11,14,16,12)(17,20,22,21)(18,19,24,23);(1,2)(3,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24);(1,4,5,6)(2,7,8,3)(9,12)(10,11)(13,14)(15,16)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947628);