Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139964 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #18162 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 756 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:
(1,2,5,10,6,12,4,8)(3,14,17,9,18,26,16,27)(7,13,23,11,20,30,21,24)(15,25,22,28,32,29,19,31);(2,8,12,10)(3,15)(9,25,27,29)(11,13,24,30)(14,31,26,28)(16,19)(17,22)(18,32);(1,3)(2,9)(4,19)(5,22)(6,18)(7,15)(8,14)(10,26)(11,29)(12,27)(13,28)(16,21)(17,23)(20,32)(24,25)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72233366, #72233489, #72233490, … |
| maximal normal subgroup of | #72233366, #72233489, #72233490, … |
| maximal quotient of | #72338940, #72338943 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47222);