Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | Aut(G) = #72306287 (order 4096) · Inn(G) = C2×#102 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,7,8,5)(4,6)(9,10,13,19,15,21,14,20)(11,22,25,38,27,39,26,36)(12,28,30,18,32,37,31,17)(16,29,33,24,35,40,34,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,29)(13,26)(14,25)(15,27)(17,36)(18,38)(19,34)(20,33)(21,35)(22,37)(23,30)(24,31)(28,39)(32,40);(3,8)(5,7)(9,12,13,30,15,32,14,31)(10,17,20,37,21,18,19,28)(11,23,25,29,27,24,26,40)(16,22,34,36,35,39,33,38)Polycyclic presentation:
2^8|a^2=e,c^2=gh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130202, #130203, #130250, … |
| maximal normal subgroup of | #130202, #130203, #130250, … |
| maximal quotient of | #122736, #122737, #122738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4881);