Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 666 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,6,26,5,10)(3,14,17,9,27,29,15,23)(4,16,22,24,21,18,20,28)(8,11,30,32,19,13,25,31);(1,3)(2,9)(4,13)(5,24,25,17)(6,27)(7,28,30,15)(8,16)(10,29,31,20)(11,21)(12,14,32,22)(18,19)(23,26);(1,4,19,9)(2,3,13,18)(5,14,30,20)(6,21,8,23)(7,29,25,22)(10,24,32,15)(11,16,26,27)(12,28,31,17)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gh,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72238636, #72238638, #72238640, … |
| maximal normal subgroup of | #72238636, #72238638, #72238640, … |
| maximal quotient of | #72267646 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46692);