Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#120 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,16,19,17,15,18,20);(5,7)(6,9)(8,12)(10,11)(13,15,16,20,17,14,18,19);(2,4);(7,12)(9,11)(14,15)(19,20);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72282406, #72283480, #72304457, … |
| maximal normal subgroup of | #72282406, #72283480, #72304457, … |
| maximal quotient of | #72305127, #72306227, C22⋊C8×#120 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294890);