Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629007 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,17)(15,19)(18,20);(5,7,8,11)(6,9,10,12)(13,14,15,17,16,18,19,20);(6,10)(9,12);(2,4)(13,14,15,17,16,18,19,20);(14,18)(17,20)Polycyclic presentation:
2^9|b^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72305406, #72305412, #123×AGL(1,ℤ/8), … |
| maximal normal subgroup of | #72305406, #72305412, #123×AGL(1,ℤ/8), … |
| maximal quotient of | #72306174, #72306204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536965);