Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484181 |
| 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 | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1425 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,23)(19,21)(22,24);(2,5)(3,8)(6,7)(12,20)(19,24);(10,14)(11,18)(12,20)(15,22)(16,17)(21,23);(9,11,13,18)(10,15,17,23)(12,19,20,24)(14,21,16,22)Polycyclic presentation:
2^9|d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338289, #72339642 |
| maximal normal subgroup of | #72338289, #72339642 |
| maximal quotient of | #72338271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391379);