Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3151 | C22 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 446 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:
(1,2,5,12,7,13,24,30)(3,14,18,25,15,29,23,27)(4,19,21,11,22,10,6,9)(8,17,26,16,28,32,31,20);(1,3,6,17,7,15,21,32)(2,8,11,27,13,28,9,25)(4,20,24,18,22,16,5,23)(10,29,30,26,19,14,12,31);(2,9)(3,15)(4,5)(6,21)(8,25)(10,19)(11,13)(14,29)(16,23)(18,20)(22,24)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=egh,f^b=fi,h^b=hi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59818);