Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(26,30)(27,32)(28,31);(1,3,6,4)(2,7,5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,26,31,32,29,27,28,30);(1,4,6,3)(2,7,5,8)(9,12,13,11)(10,14,15,16)(19,23)(21,24)(25,27,29,26)(28,30,31,32);(1,5,6,2)(3,8,4,7)(25,27,28,32,29,26,31,30)Polycyclic presentation:
2^9|a^2=egi,b^2=efh,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378119);