Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141701 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33717 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 426 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:
(1,2)(3,7)(4,8)(5,6)(10,17,20,23)(11,21,19,22)(12,18)(13,24)(14,15);(1,3,7,2,5,8,4,6)(9,10,18,19,15,22,12,23,16,20,13,11,14,21,24,17);(2,6)(3,8)(9,11,18,22,15,17,12,20,16,19,13,21,14,23,24,10)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eghi,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309412, #72309430, #72309431, … |
| maximal normal subgroup of | #72309412, #72309430, #72309431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48936);