Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C24 |
| commutator, socle | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | #3481 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1383 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)(3,7)(4,6)(5,8)(9,10,13,17,14,18,24,30,12,15,22,26,23,27,34,37)(11,16,20,28,21,29,33,38,19,25,31,35,32,36,39,40);(1,3)(2,5)(4,8)(6,7)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(3,8)(5,7);(2,6)(5,7)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(9,11,13,20,14,21,24,33,12,19,22,31,23,32,34,39)(10,16,17,28,18,29,30,38,15,25,26,35,27,36,37,40)Polycyclic presentation:
2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cfi200 conjugacy classes — view the full character table.
| maximal subgroup of | #72287023, #72287764, #72287798, … |
| maximal normal subgroup of | #72287023, #72287764, #72287798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529954);