Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(4,5)(7,8)(9,10,13,18,15,20,21,24,14,19,12,23,11,17,22,16);(1,2,4,3,6,8,5,7)(10,16,19,24)(11,15)(12,22)(13,21)(17,18,20,23);(1,3,4,8,6,7,5,2)(9,11,14,15)(10,17,19,20)(12,21,13,22)(16,23,24,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=eg,d^2=ghi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cg,d^b=dh,f^b=fh,h^b=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309855, #72309861, #72309862, … |
| maximal normal subgroup of | #72309855, #72309861, #72309862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48503);