Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60741 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,18,13,28)(11,16,22,32,23,31,21,38)(12,24,26,17,27,35,29,34)(20,36,37,30,25,40,39,33);(1,3)(2,5)(4,8)(6,7)(9,11,14,22,15,23,13,21)(10,16,19,32,18,31,28,38)(12,25,26,39,27,20,29,37)(17,33,35,36,34,30,24,40);(9,12,14,26,15,27,13,29)(10,17,19,35,18,34,28,24)(11,20,22,37,23,25,21,39)(16,30,32,40,31,33,38,36);(3,8)(5,7)(10,18)(16,31)(17,34)(19,28)(24,35)(30,33)(32,38)(36,40);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301869);