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 | |
5508 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)(3,5)(4,7)(6,8)(18,20)(22,24);(1,3,6,4)(2,7,8,5)(9,10)(11,14)(12,13)(15,16)(19,23)(22,24);(2,8)(5,7)(10,13)(14,16)(17,18,21,20)(19,22,23,24);(2,8)(3,4)(9,11,12,15)(10,14,13,16)(17,19,21,23)(18,22,20,24);(1,4,6,3)(2,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22);(1,5)(2,3)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8196475);