Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | #61141 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1760 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)(9,10,14,18,15,19,13,28)(11,20,24,29,21,33,23,16)(12,17,27,32,25,31,26,38)(22,34,37,39,35,40,36,30);(1,3)(2,5)(4,8)(6,7)(11,21)(12,25)(16,29)(17,31)(20,33)(23,24)(26,27)(32,38);(2,6)(5,7)(9,11,13,23,15,21,14,24)(10,16,18,20,19,29,28,33)(12,22,26,36,25,35,27,37)(17,30,32,34,31,39,38,40);(3,8)(5,7)(12,25)(17,31)(22,35)(26,27)(30,39)(32,38)(34,40)(36,37);(2,6)(5,7)(9,12,14,27,15,25,13,26)(10,17,18,32,19,31,28,38)(11,22,24,37,21,35,23,36)(16,30,20,34,29,39,33,40)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311155);