Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(13,14)(15,21)(16,20)(17,19)(18,22)(23,28)(24,26)(25,27);(5,6,7,9,8,10,11,12)(13,15)(14,18)(16,24,25,22)(17,23)(19,26)(20,28,27,21);(1,3,4,2)(14,19)(18,26)(20,27)(21,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,f^2=h,b^a=bd,c^a=ci,f^a=fg,d^b=dg,g^b=gi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35502);