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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1068 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)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,26)(27,32)(28,38)(29,39)(30,34)(31,36)(33,35)(37,40);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,19,20,23)(18,21,22,24)(25,27,35,28,30,37,29,36)(26,31,39,40,34,38,33,32);(1,3)(2,6)(4,8)(5,7)(10,15)(14,16)(19,23)(21,24)(27,28)(29,35)(31,40)(32,38)(33,39)(36,37);(3,8)(6,7)(9,12,13,11)(10,16,15,14)(25,28,29,27,30,36,35,37)(26,32,33,38,34,40,39,31)Polycyclic presentation:
2^9|a^2=egi,b^2=efgh,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, 381467);