Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141701 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,25,28,38)(12,31,27,36)(13,21)(14,20)(15,23)(16,22)(17,24)(18,26,35,32)(19,29,37,30)(33,40)(34,39);(1,3,7,2,5,8,4,6)(9,11,14,12,16,30,13,32,17,28,33,27,15,29,34,26)(10,18,21,37,23,31,20,38,24,35,39,19,22,36,40,25);(2,6)(3,8)(9,12,14,29,16,32,13,11,17,27,33,30,15,26,34,28)(10,19,21,31,23,25,20,35,24,37,39,36,22,38,40,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egh,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309414, #72309427, #72309431, … |
| maximal normal subgroup of | #72309414, #72309427, #72309431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48935);