Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 873 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,4)(5,6)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22);(1,3,7,6)(2,5,8,4)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=h,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72232758, #72232828, #72232832, … |
| maximal normal subgroup of | #72232758, #72232828, #72232832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47046);