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 | #18164 | 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 | |
662 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 662 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:
(1,2,3,12,16,8,20,10)(4,18,6,13,17,29,7,14)(5,9,19,26,21,11,30,25)(15,28,22,24,32,31,23,27);(1,3,6,17)(2,8)(4,16,20,7)(5,15,22,30)(9,24)(10,18)(11,27)(12,29)(13,14)(19,21,32,23)(25,26)(28,31);(2,9)(3,4)(5,21)(8,11)(10,25)(12,26)(13,24)(14,27)(15,30)(17,20)(18,28)(19,32)(22,23)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,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 | #72231376, #72233827, #72233829, … |
| maximal normal subgroup of | #72231376, #72233827, #72233829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47236);