Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,19,16);(6,10,12,11)(13,15,18,14,17,19,16,20);(13,16,17,18)(14,15,20,19);(5,7,9,8)(6,11,12,10)(13,16,17,18)(14,15,20,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=h,g^2=i,b^a=be,e^a=eh,g^a=ghi,e^b=ei152 conjugacy classes — view the full character table.
| maximal subgroup of | #72209971, #72209972, #72238823, … |
| maximal normal subgroup of | #72209971, #72209972, #72238823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88752);