Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347226 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,7)(4,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(1,2,5,3)(4,7,8,6)(12,21)(15,20)(17,24)(19,22);(9,11)(10,14)(12,22)(13,18)(15,24)(16,23)(17,20)(19,21);(1,3,5,2)(4,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254477);