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 | |
4881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,5,7)(9,10,14,13)(11,16,15,12)(17,18,21,22)(19,20,23,24);(1,3,6,5)(2,7,4,8)(9,11,14,15)(10,12,13,16)(18,22)(19,23);(1,3,6,5)(2,8,4,7)(9,11,14,15)(10,12,13,16)(17,19)(18,20)(21,23)(22,24);(1,2,6,4)(3,8,5,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23);(1,4,6,2)(3,8,5,7)(9,13,14,10)(11,12,15,16);(1,5,6,3)(2,7,4,8)(9,12)(10,15)(11,13)(14,16)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993569);