Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142969 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #18162 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 851 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 24 points:
(3,8)(9,10,14,17)(11,18,20,23)(12,22,13,16)(15,21,24,19);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,23)(17,24)(18,22);(2,5)(3,8)(6,7)(11,19)(12,13)(15,23)(16,22)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=g,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,h^b=hi,d^c=dh,f^c=fgi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72231834, #72235671, #72235685, … |
| maximal normal subgroup of | #72231834, #72235671, #72235685, … |
| maximal quotient of | #72237824, #72237828, #72237850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50178);