Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C2×C8 |
| Frattini, cosocle | C2×#322 | C22 |
| socle | C22 | #1271 |
| prime radical | C23 | D4⋊C8 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×#322 ▷ C2×C42 ▷ C22 ▷ C1 | |
155 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,7,12,14,5,10,13,16,3,8,11,15)(18,22,23,30,25,31,24,26);(2,7,10,8)(9,15,16,14)(17,18)(19,26)(20,23)(21,25)(22,28)(24,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fhi,d^2=g,f^2=i,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,f^b=fi,g^b=gi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 74);