Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485324 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #18164 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1133 conjugacy classes of subgroups — too many to draw. Only its 124 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)(4,6)(7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(2,5)(3,8)(6,7)(15,20)(17,23);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(2,6)(5,7)Polycyclic presentation:
2^9|c^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231054, #72240247, #72278246, … |
| maximal normal subgroup of | #72231054, #72240247, #72278246, … |
| maximal quotient of | #72240575, #72278636, #72279195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392522);