Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #25743 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,17)(11,19,21,15)(12,23,25,38)(14,28,27,18)(16,31,32,26)(20,35,24,39)(22,30,37,34)(29,36,33,40);(2,5)(6,7)(9,11,13,21)(10,15,17,19)(12,24,25,20)(14,22,27,37)(16,29,32,33)(18,30,28,34)(23,35,38,39)(26,36,31,40);(9,12,14,26,13,25,27,31)(10,16,18,23,17,32,28,38)(11,20,22,36,21,24,37,40)(15,29,30,39,19,33,34,35);(1,3,4,8)(2,6,5,7)(10,17)(11,21)(16,32)(18,28)(20,24)(22,37)(23,38)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72311085, #72311099, #72311177, … |
| maximal normal subgroup of | #72311085, #72311099, #72311177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410033);