Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94658 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7408 |
| commutator | #423 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| prime radical | #10212 | C2 |
| LCS | G ▷ #423 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10)(3,14,16,27)(5,12,20,28)(6,13,22,30)(7,11,21,29)(8,19,23,17)(9,25,26,18)(15,24,31,32);(2,8)(3,15)(5,6)(9,12)(10,27)(11,24)(13,14)(16,19)(17,18)(20,22)(23,28)(25,31)(26,29)(30,32)Polycyclic presentation:
2^9|a^2=d,c^2=fi,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72338336, #72338410, #72338411, … |
| maximal normal subgroup of | #72338336, #72338410, #72338411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1854);