Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #13047 |
| prime radical | #18984 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 487 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:
(1,2,7,12,8,14,24,31)(3,15,18,4,10,22,29,23)(5,11,16,13,25,30,6,26)(9,27,20,28,21,19,32,17);(1,3,16,27,8,10,6,19)(2,9,13,4,14,21,26,23)(5,17,7,18,25,28,24,29)(11,15,12,20,30,22,31,32);(1,4)(2,10)(3,14)(5,15)(6,9)(7,20)(8,23)(11,29)(12,17)(13,19)(16,21)(18,30)(22,25)(24,32)(26,27)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,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=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59775);