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 | #35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1954 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1954 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,20,11,16)(12,17,22,23,13,18,21,24);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,20,19)(12,22,13,21)(17,23,18,24);(3,8)(5,7)(10,17)(12,13)(16,23)(18,20)(19,24)(21,22);(2,6)(5,7)(9,12,15,13)(10,17,20,18)(11,21,14,22)(16,24,19,23)Polycyclic presentation:
2^9|a^2=g,b^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.
| maximal subgroup of | #72213335, #72213400, #72213406, … |
| maximal normal subgroup of | #72213335, #72213400, #72213406, … |
| maximal quotient of | #72235020, #72235151, #72235261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403418);