Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1742 conjugacy classes of subgroups — too many to draw. Only its 206 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 — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,20,16,21,11,17)(12,24,22,38,27,37,30,40)(13,19,28,33,14,29,23,32)(18,34,35,25,36,39,31,26);(1,3)(2,5)(4,8)(6,7)(9,11,16,15)(10,17,21,20)(12,22,27,30)(13,28,14,23)(18,31,36,35)(19,33,29,32)(24,38,37,40)(25,34,26,39);(3,8)(5,7)(9,12,16,27)(10,18,21,36)(11,22,15,30)(13,26,14,25)(17,31,20,35)(19,37,29,24)(23,34,28,39)(32,38,33,40);(2,6)(5,7)(9,13,16,14)(10,19,21,29)(11,23,15,28)(12,25,27,26)(17,32,20,33)(18,24,36,37)(22,39,30,34)(31,40,35,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227698, #72227734, #72227736, … |
| maximal normal subgroup of | #72227698, #72227734, #72227736, … |
| maximal quotient of | #72268016, #72268113, #72268321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403420);