Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139809 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9497 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 683 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:
(1,2,6,10,7,12,4,18)(3,13,16,25,17,30,14,8)(5,11,19,27,22,9,21,26)(15,29,31,24,20,28,32,23);(1,3)(2,8,12,25)(4,16,6,14)(5,20)(7,17)(9,23,11,24)(10,13)(15,22)(18,30)(19,31,21,32)(26,29)(27,28);(2,9,12,11)(8,23,25,24)(10,27,18,26)(13,28,30,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72302897, #72302902, #72302905, … |
| maximal normal subgroup of | #72302897, #72302902, #72302905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47007);