Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501120 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 148 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)(11,19,21,36)(13,25,23,17)(14,27,24,18)(16,30,29,22)(20,37,38,35)(26,33,28,34)(31,40,32,39);(1,3,5,4)(2,6,8,7)(10,15)(11,20)(14,24)(16,31)(17,25)(19,35)(21,38)(22,40)(26,28)(29,32)(30,39)(36,37);(9,11,13,22,12,21,23,30)(10,16,17,19,15,29,25,36)(14,20,28,39,24,38,26,40)(18,32,34,35,27,31,33,37);(1,4,5,3)(2,7,8,6)(10,15)(16,29)(17,25)(18,27)(19,36)(31,32)(33,34)(35,37)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=f,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,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410560);