Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499873 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1552 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 1552 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,18,17,14)(12,19,21,24)(16,22,23,20);(1,3,4,8)(2,5,6,7)(9,11,13,17)(10,14,15,18)(12,20,21,22)(16,19,23,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=i,b^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.
| maximal subgroup of | #72213168, #72267418, #72267434, … |
| maximal normal subgroup of | #72213168, #72267418, #72267434, … |
| maximal quotient of | #72213149, #72267372, #72267377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407225);