Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213555 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
927 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 927 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16,14,17,12,20)(11,15,19,24,18,22,21,23)(25,26,28,33,29,32,37,40)(27,30,35,38,34,31,36,39);(1,3)(2,5)(4,8)(6,7)(11,18)(15,22)(19,21)(23,24)(25,27,29,34)(26,30,32,31)(28,35,37,36)(33,38,40,39);(3,8)(5,7)(9,11)(10,15)(12,21)(13,19)(14,18)(16,24)(17,22)(20,23)(25,27)(26,31)(28,36)(29,34)(30,32)(33,38)(35,37)(39,40);(2,6)(5,7)(9,12,14,13)(10,16,17,20)(11,19,18,21)(15,23,22,24)(26,32)(27,34)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120759);