Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152590 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13047 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #19052 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,5)(4,14,18,32)(6,23,22,15)(7,11,25,24)(8,12)(9,17,27,31)(10,13)(16,28,21,29)(19,26,20,30);(1,3,13,12,8,10,5,2)(4,14,31,29,20,30,21,9)(6,24,25,23,22,11,7,15)(16,27,18,32,17,28,19,26);(1,4)(2,9)(3,14)(5,21)(6,17)(7,18)(8,20)(10,30)(11,27)(12,29)(13,31)(15,32)(16,22)(19,25)(23,26)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,e^c=ehi,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59787);