Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139736 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23915 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
740 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 740 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:
(1,2,6,11)(3,12)(4,9)(5,10)(7,16,24,13)(8,20)(14,29,19,28)(15,30)(17,25,21,26)(18,27)(22,32)(23,31);(1,3,4,16)(2,7,9,12)(5,19,17,31)(6,15,18,13)(8,22,26,28)(10,23,25,29)(11,24,27,30)(14,21,32,20);(2,8)(5,20)(7,22)(9,26)(10,11)(12,28)(14,19)(17,21)(23,24)(25,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dgi,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72238016, #72303148, #72303154, … |
| maximal normal subgroup of | #72238016, #72303148, #72303154, … |
| maximal quotient of | C2×#139736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46932);