Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,9,23,10,29,20)(3,13,15,25,16,21,24,32)(4,19,14,17,28,8,27,18)(5,11,22,30,7,12,26,31);(1,3,14,10,5,17,32,12)(2,7,18,25,11,23,16,27)(4,20,6,24,21,31,22,19)(8,28,9,29,15,13,30,26);(4,21)(6,22)(8,15)(9,30)(13,28)(19,24)(20,31)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237486, #72238433, #72238435, … |
| maximal normal subgroup of | #72237486, #72238433, #72238435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47106);