Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150584 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,17,14,19,25,34)(11,20,22,37,21,35,38,33)(13,16,28,32,29,18,26,36)(15,24,30,27,31,39,40,23);(2,5)(3,8)(6,7)(9,11)(10,15,19,31)(12,24)(13,27)(14,21)(16,20,18,35)(17,33,34,37)(22,28)(23,29)(25,39)(26,38)(30,36,40,32);(2,6)(5,7)(12,25)(17,34)(22,38)(24,39)(26,28)(30,40)(32,36)(33,37)Polycyclic presentation:
2^9|a^2=e,b^2=h,d^2=gi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efg,f^b=fi,e^c=ei,e^d=ehi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72292960, #72292970, #72293065, … |
| maximal normal subgroup of | #72292960, #72292970, #72293065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57784);