Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153545 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13584 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | #19745 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10,14,6)(4,15,13,9)(5,8)(7,11,16,12);(2,6)(3,11)(4,13)(7,15)(8,16)(9,10)(12,14);(3,12)(6,16)(7,10)(11,14)Polycyclic presentation:
2^9|a^2=e,d^2=ghi,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dgh,e^b=efgi,f^b=fhi,g^b=gi,d^c=di,e^d=egh,f^e=fhi,g^e=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72231248, #72231261, #72231262, … |
| maximal normal subgroup of | #72231248, #72231261, #72231262, … |
| maximal quotient of | C2×#153545, #72283874, #72309199 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60741);