Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162550 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1094 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(11,14,15,16)(17,18,21,24,22,19,20,23);(1,3,4,8)(2,5,6,7)(9,10,13,12)(11,14,15,16);(3,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23);(2,6)(5,7)(9,10,13,12)(11,14,15,16)(17,20,22,21)(18,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=ehi,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fi,g^c=ghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239243, #72239264 |
| maximal normal subgroup of | #72239243, #72239264 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122378);