Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,21,23,24)(25,26)(27,36)(28,33)(29,32)(30,39)(31,35)(34,37)(38,40);(2,5)(6,8)(10,13)(14,15)(17,19)(18,21)(20,23)(22,24)(25,27,37,38,31,28,30,29)(26,32,39,33,35,40,34,36);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(19,23)(21,24)(25,28,31,27)(26,33,35,36)(29,37,38,30)(32,34,40,39);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(25,29,30,28,31,38,37,27)(26,33,34,32,35,36,39,40)Polycyclic presentation:
2^9|a^2=egi,b^2=hi,c^2=i,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378228);