Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
749 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(2,5,6,7)(9,10,13,18,14,17,26,31)(11,15,21,28,20,29,36,32)(12,22,24,33,23,16,25,30)(19,27,35,39,34,38,37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15,17,29)(12,19)(13,21)(14,20)(16,27,22,38)(18,32,31,28)(23,34)(24,35)(25,37)(26,36)(30,40,33,39);(9,12,14,23)(10,16,17,22)(11,19,20,34)(13,25,26,24)(15,27,29,38)(18,33,31,30)(21,37,36,35)(28,40,32,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72266604 |
| maximal normal subgroup of | #72266604 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37020);