Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | |
532 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 532 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:
(1,2,3,12,17,30,4,19)(5,21,23,26,32,11,22,29)(6,13,18,8,7,14,15,10)(9,24,28,20,31,25,27,16);(2,8,13,12)(3,15)(4,18)(9,27,29,11)(10,14,19,30)(16,23)(20,22)(21,31,28,26);(1,3,6,18)(2,9,13,29)(4,7,15,17)(5,20,24,23)(8,21,12,28)(10,11,19,27)(14,26,30,31)(16,25,22,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,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 | #72277922 |
| maximal normal subgroup of | #72277922 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47260);