Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(14,18,20,19);(5,7,9,8)(6,10,12,11)(14,19,20,18);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,14)(15,19)(16,18)(17,20);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72215799, #72216040, #72216045, … |
| maximal normal subgroup of | #72215799, #72216040, #72216045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276541);