Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141233 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | #19603 | C2 |
| 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 | |
393 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 102 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)(9,10,11,17,13,18,19,14,12,15,20,16,21,23,24,22);(1,2,5,7,6,8,4,3)(10,14,15,22)(11,19)(13,21)(16,18,17,23)(20,24);(1,3,4,8,6,7,5,2)(10,15)(14,22)(16,17)(18,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,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,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72309409, #72309415, #72309416, … |
| maximal normal subgroup of | #72309409, #72309415, #72309416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48446);