Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #390 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,21,24)(19,22,27,26)(20,23)(29,30)(31,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,21,27)(18,22,24,26)(20,28,29,25)(23,32,30,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,18,23,21,29,24,30)(19,25,26,32,27,28,22,31)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,c^b=ch,d^b=df,e^b=eghi,g^b=gi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58667);