Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480684 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1090 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)(11,19,22,34)(12,16)(13,26,28,17)(14,18)(15,21)(20,35,23,37)(24,39,40,32)(25,33)(27,30)(29,36)(31,38);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11,14,22)(10,15,18,30)(12,23,25,20)(13,27,28,21)(16,31,33,29)(17,34,26,19)(24,36,40,38)(32,35,39,37);(2,6)(5,7)(9,12,14,25)(10,16,18,33)(11,20,22,23)(13,24,28,40)(15,29,30,31)(17,32,26,39)(19,35,34,37)(21,36,27,38)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311332, #72311343, #72311370, … |
| maximal normal subgroup of | #72311332, #72311343, #72311370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403509);