Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139747 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
743 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 743 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,11,15,26)(4,16)(5,19,12,9)(6,10)(7,14)(8,17,24,27)(13,21)(18,23)(20,22)(25,28,32,29)(30,31);(1,3,12,14,6,15,5,20)(2,7,19,26,10,22,9,11)(4,13,27,29,18,30,17,28)(8,21,23,32,24,31,16,25);(1,4)(2,8)(3,13)(5,17)(6,18)(7,21)(9,16)(10,24)(11,25)(12,27)(14,29)(15,30)(19,23)(20,28)(22,31)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72240474, #72240482, #72255764, … |
| maximal normal subgroup of | #72240474, #72240482, #72255764, … |
| maximal quotient of | #72337627, #72337649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46943);