Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | #3492 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,3,12,16,29,4,19)(5,21,24,9,27,30,23,32)(6,13,18,8,7,14,15,10)(11,20,31,22,28,17,26,25);(2,8,13,12)(3,15)(4,18)(5,22)(9,28,31,30)(10,14,19,29)(11,26,21,32)(25,27);(1,3,16,4)(2,9,29,32)(5,20,27,17)(6,18,7,15)(8,21,10,30)(11,19,28,12)(13,31,14,26)(22,23,25,24)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gi,d^2=gi,e^2=gi,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72324327, #72324331, #72324339 |
| maximal normal subgroup of | #72324327, #72324331, #72324339 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47261);