Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139604 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #23948 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,3,12,5,8,16,10)(4,18,21,27,22,9,26,28)(6,24,14,29,7,13,17,32)(11,19,31,20,25,15,30,23);(1,3,5,16)(2,8)(4,19,22,15)(6,14,7,17)(9,11)(10,12)(13,24)(18,25)(20,26,23,21)(27,30)(28,31)(29,32);(2,9)(3,14)(4,20)(8,18)(10,28)(11,13)(12,27)(15,19)(16,17)(21,26)(22,23)(24,25)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=cgi,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251721, #72251999, #72252006 |
| maximal normal subgroup of | #72251721, #72251999, #72252006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46845);