Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
844 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 844 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(1,2,5,8)(3,7,4,6)(9,10,12,15)(11,19,21,34)(13,25,23,17)(14,28,24,18)(16,26,31,22)(20,38,29,37)(27,35,30,36)(32,40,33,39);(1,3,5,4)(2,6,8,7)(10,15)(11,20)(14,24)(16,32)(17,25)(19,37)(21,29)(22,40)(26,39)(27,30)(31,33)(34,38);(9,11)(10,16)(12,21)(13,26)(14,29)(15,31)(17,34)(18,32)(19,25)(20,24)(22,23)(27,40)(28,33)(30,39)(35,38)(36,37);(1,4,5,3)(2,7,8,6)(10,15)(16,31)(17,25)(18,28)(19,34)(32,33)(35,36)(37,38)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408168);