Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111358 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
964 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,14,19,15,17,13,18)(11,20,16,23,21,12,22,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,22)(15,21)(17,20)(18,24)(19,23);(1,2)(3,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,20)(17,22)(18,21)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dhi,e^c=ei,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237274, #72237275, #72239660, … |
| maximal normal subgroup of | #72237274, #72237275, #72239660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18555);