Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127185 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,23,24,16)(13,17)(14,18)(15,21,27,33)(20,31)(22,32)(25,36,37,29)(26,30)(28,35,38,40)(34,39);(2,5,7,6)(9,11,13,22,14,20,26,34)(10,15,17,28,18,27,30,38)(12,21,25,35,24,33,37,40)(16,19,29,32,23,31,36,39);(11,20)(15,27)(19,31)(21,33)(22,34)(28,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gh,f^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ci,e^b=eh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284436, #72284458, #72284501, … |
| maximal normal subgroup of | #72284436, #72284458, #72284501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34385);