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 | |
5196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,15)(11,13,12,16)(17,18)(19,23)(20,21)(22,24);(1,2,5,4)(3,6,7,8)(10,15)(11,12)(18,21)(22,23);(2,4)(6,8)(9,11,14,12)(10,16,15,13)(17,18)(19,22)(20,21)(23,24);(3,7)(6,8)(9,12,14,11)(10,13,15,16)(17,19)(18,22)(20,24)(21,23);(1,3,5,7)(2,6,4,8)(9,13)(10,11)(12,15)(14,16);(1,4,5,2)(3,8,7,6)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=h,e^2=gi,f^2=gi,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, 8249009);