Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94666 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7408 |
| commutator | #423 | C2×C4 |
| Frattini, cosocle | #1962 | C22 |
| prime radical | #532 | D4 |
| LCS | G ▷ #423 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1962 ◁ G | |
| LECS | G ▷ #1962 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 181 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:
(1,2,4,10,5,13,21,27)(3,14,15,25,16,29,31,8)(6,11,20,28,7,12,22,9)(17,30,19,24,18,26,32,23);(1,3,5,16)(2,8,13,25)(4,19,21,32)(6,18,7,17)(9,26,28,30)(10,29,27,14)(11,24,12,23)(15,22,31,20)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fi,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=ghi,d^c=dghi,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1865);