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 | |
5428 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)(3,8)(4,6)(5,7)(10,12)(14,16)(17,18,21,23)(19,20,24,22);(1,3,5,6)(2,4,7,8)(11,15)(14,16)(17,19,21,24)(18,22,23,20);(1,4)(2,6)(3,7)(5,8)(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,24)(22,23);(3,6)(4,8)(17,20)(18,19)(21,22)(23,24);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15);(1,2)(3,4)(5,7)(6,8)(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841551);