Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131979 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #52083 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#536 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
800 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,19,20,32)(12,23,25,15)(13,17)(14,18)(16,22,31,26)(21,34)(24,38,40,29)(27,33)(28,37)(30,39)(35,36);(1,2)(3,7)(4,6)(5,8)(10,15)(11,20)(16,30)(17,29)(18,23)(19,34)(21,36)(22,26)(28,31)(32,35)(33,38)(37,39);(3,8)(5,7)(9,11,14,20)(10,16,18,31)(12,21,25,36)(13,26,27,22)(15,28,23,30)(17,32,33,19)(24,39,40,37)(29,34,38,35)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72283955, #72285866, #72318049, … |
| maximal normal subgroup of | #72283955, #72285866, #72318049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39179);