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 | |
765 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 765 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:
(1,2,5,10,6,9,19,20)(3,11,14,27,13,7,18,22)(4,15,16,25,17,8,23,24)(12,21,29,31,28,26,30,32);(1,3,5,18)(2,7,20,27)(4,12,16,29)(6,13,19,14)(8,21,24,32)(9,11,10,22)(15,26,25,31)(17,28,23,30);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,15)(10,25)(11,26)(13,28)(14,30)(18,29)(19,23)(20,24)(22,31)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=i,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 | #72233445, #72233454, #72233461, … |
| maximal normal subgroup of | #72233445, #72233454, #72233461, … |
| maximal quotient of | #72246453, #72253662, #72266325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46945);