Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
727 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 727 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,13,17,14,11,15,19)(12,18,21,24,20,16,22,23);(1,3,4,6)(2,5,8,7)(9,11,13,19,14,10,15,17)(12,18,22,23,20,16,21,24);(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,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 | #72303903, #72304158, #72304159 |
| maximal normal subgroup of | #72303903, #72304158, #72304159 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47126);