Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 | #60673 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,29)(11,16,23,32,21,31,22,38)(12,24,27,34,28,40,26,17)(20,35,37,33,25,39,36,30);(2,5)(3,7)(9,11,14,23,15,21,13,22)(10,16,18,32,19,31,29,38)(12,25,27,36,28,20,26,37)(17,33,24,39,34,30,40,35);(9,12,13,26,15,28,14,27)(10,17,18,24,19,34,29,40)(11,20,22,36,21,25,23,37)(16,30,32,35,31,33,38,39);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(11,21)(16,31)(20,25)(22,23)(30,33)(32,38)(35,39)(36,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,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=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313275, #72313284, #72313312, … |
| maximal normal subgroup of | #72313275, #72313284, #72313312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296578);