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 | |
5287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,8,4,6)(17,18)(19,23)(20,22)(21,24);(3,4)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18,20,22)(19,21,24,23);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24);(3,4)(6,8)(9,10)(11,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,16)(13,15);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9994914);