Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12708 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #535 | C23 |
| prime radical | #3508 | C22 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3508 ◁ G | |
| LECS | G ▷ #535 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 564 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:
(1,2,6,12,7,14,5,11)(3,15,18,31,19,27,32,26)(4,10,21,30,22,13,24,9)(8,23,25,16,28,17,29,20);(1,3,7,19)(2,8,14,28)(4,20,22,16)(5,23,6,17)(9,29,30,25)(10,27,13,15)(11,26,12,31)(18,24,32,21);(1,4,7,22)(2,9)(3,16,19,20)(5,21,6,24)(8,25)(10,11)(12,13)(14,30)(15,26)(17,18,23,32)(27,31)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59352);