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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1648 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)(9,10,16,19,14,20,13,27)(11,22,23,31,12,17,25,33)(15,21,28,39,29,35,30,38)(18,34,36,26,32,40,37,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,14)(13,25)(15,24)(16,23)(18,35)(19,33)(20,22)(21,32)(26,29)(27,31)(28,34)(30,40)(36,38)(37,39);(3,8)(5,7)(9,12,14,11)(10,18,20,32)(13,23,16,25)(15,24,29,26)(17,21,22,35)(19,37,27,36)(28,34,30,40)(31,39,33,38);(2,6)(5,7)(9,13,14,16)(10,19,20,27)(11,23,12,25)(15,28,29,30)(17,31,22,33)(18,36,32,37)(21,38,35,39)(24,40,26,34)Polycyclic presentation:
2^9|a^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403419);