Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,19,13,17,15,30)(11,20,23,37,22,36,29,38)(12,24,27,32,26,40,28,18)(16,31,34,21,33,39,35,25);(2,6)(5,8)(9,11,12,25,15,23,28,31,13,22,26,21,14,29,27,39)(10,16,32,20,19,35,40,38,17,33,18,36,30,34,24,37);(1,2,3,5,4,6,7,8)(10,17)(16,33)(18,32)(19,30)(20,36)(24,40)(34,35)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224822, #72224823, #72224824, … |
| maximal normal subgroup of | #72224822, #72224823, #72224824, … |
| maximal quotient of | #72226938, #72226940 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48951);