Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,13,17,14,18,26,33)(11,19,21,34,22,30,35,39)(12,16,24,31,25,32,37,40)(15,27,28,23,29,36,38,20);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,23,25,20)(13,21)(14,22)(16,30,32,19)(17,28)(18,29)(24,36,37,27)(26,35)(31,39,40,34)(33,38);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh,h^b=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264419, #72264458, #72264463, … |
| maximal normal subgroup of | #72264419, #72264458, #72264463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34765);