Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
770 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,6,10,5,11,21,29)(3,13,9,27,16,4,18,14)(7,25,23,30,8,12,24,15)(17,19,28,31,26,20,32,22);(1,3,14,25,8,17,22,11)(2,5,16,27,12,7,26,31)(4,15,21,18,20,10,23,32)(6,9,19,29,24,28,13,30);(4,13)(6,21)(9,18)(10,29)(15,30)(19,20)(23,24)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72233278, #72233279, #72233283, … |
| maximal normal subgroup of | #72233278, #72233279, #72233283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46676);