Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,16,19)(15,17,20,18);(1,3,4,2)(5,7)(6,9)(8,12)(10,11);(14,17)(15,20)(18,19);(7,12)(9,11);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=h,b^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi122 conjugacy classes — view the full character table.
| maximal subgroup of | #72282132, #72282262, #72305266, … |
| maximal normal subgroup of | #72282132, #72282262, #72305266, … |
| maximal quotient of | #72305835, #72306197, #72310240 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294781);