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 | |
5902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,13)(12,15)(14,16)(17,18,22,20)(19,23,21,24);(1,2)(3,4)(5,6)(7,8)(10,11)(14,15)(17,19,22,21)(18,23,20,24);(2,6)(4,7)(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,24,20,23);(1,3)(2,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24);(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102436);