Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18,15,19,14,29)(11,20,23,36,21,34,24,39)(12,25,26,31,28,40,27,17)(16,30,33,37,32,38,35,22);(9,11,12,22,13,23,26,30,15,21,28,37,14,24,27,38)(10,16,31,34,29,35,25,36,19,32,17,20,18,33,40,39);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,21)(16,32)(20,34)(22,37)(23,24)(30,38)(33,35)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319933, #72319942, #72319959, … |
| maximal normal subgroup of | #72319933, #72319942, #72319959, … |
| maximal quotient of | C2×#495561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402757);