Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #392 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 80 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)(12,23,25,15)(13,26)(14,18)(16,21)(17,27)(20,37,39,36)(22,34)(24,31,29,35)(28,33)(30,38,32,40);(1,2)(3,7)(4,6)(5,8)(10,15)(13,27)(16,30)(17,31)(18,23)(19,36)(21,28)(24,29)(26,35)(32,33)(34,37)(38,40);(9,11,14,22)(10,16,18,33)(12,20,25,39)(13,28,27,21)(15,30,23,32)(17,34,26,19)(24,40,29,38)(31,37,35,36)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,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72284868 |
| maximal normal subgroup of | #72284868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37311);