Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 701 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:
(1,2,7,11,27,12,4,9)(3,14,18,21,19,24,32,28)(5,16,15,20,31,10,30,26)(6,25,23,29,8,13,22,17);(1,3,15,25)(2,6,26,21)(4,16,24,17)(5,11,23,32)(7,10,14,29)(8,20,28,12)(9,22,18,31)(13,27,19,30);(1,4,8,23)(2,9,13,29)(3,16,20,32)(5,15,24,28)(6,22,27,7)(10,26,18,19)(11,25,17,12)(14,21,31,30)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72234709, #72234713, #72234714, … |
| maximal normal subgroup of | #72234709, #72234713, #72234714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46681);