Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| 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 | #60682 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1864 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)(3,6)(4,5)(7,8)(9,10,14,19,15,20,28,18)(11,16,24,32,23,30,38,31)(12,25,13,17,26,33,27,34)(21,35,22,29,36,40,37,39);(2,5)(6,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,29)(18,31)(19,32)(20,30)(25,35)(26,36)(27,37)(28,38)(33,40)(34,39);(9,12,14,13,15,26,28,27)(10,17,18,25,20,34,19,33)(11,21,24,22,23,36,38,37)(16,29,31,35,30,39,32,40);(9,13,14,26,15,27,28,12)(10,17,19,33,20,34,18,25)(11,22,24,36,23,37,38,21)(16,29,32,40,30,39,31,35);(1,3,4,7)(2,6,5,8)(11,23)(16,30)(21,36)(22,37)(24,38)(29,39)(31,32)(35,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239527, #72239543, #72239829, … |
| maximal normal subgroup of | #72239527, #72239543, #72239829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281215);