Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213564 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
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 | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1044 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 210 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)(11,17,18,15)(12,20)(13,14)(16,21)(19,22,24,23)(25,26,29,33,30,34,28,32)(27,31,35,38,36,39,37,40);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(12,21)(22,23)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,38)(34,39);(3,8)(5,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23);(2,6)(5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,28,30,29)(26,32,34,33)(27,35,36,37)(31,38,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122365);