Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499873 |
| 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 | C2×2−1+4 | C23 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 862 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,19,20,24)(12,17,18,14)(16,22,23,21);(1,3,5,4)(2,6,8,7)(10,15)(11,16)(14,17)(19,21)(20,23)(22,24);(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,21)(18,24);(1,4,5,3)(2,7,8,6)(9,12,13,18)(10,17,15,14)(11,19,20,24)(16,21,23,22)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,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=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408170);