Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #2021 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 47 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)(3,12)(4,16)(5,8)(6,11)(7,13)(9,19)(10,20)(14,27)(15,25)(17,22)(18,26)(21,31)(23,29)(24,32)(28,30);(1,3)(2,7)(4,17)(5,14)(6,15)(8,23)(9,25)(10,21)(11,22)(12,16)(13,19)(18,31)(20,30)(24,28)(26,27)(29,32);(2,8)(5,20)(7,21)(9,24)(10,11)(12,27)(13,17)(14,30)(16,26)(18,32)(22,23)(25,28)Polycyclic presentation:
2^8|d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,g^b=gh,d^c=dg,e^c=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13353, #498940, #498941, … |
| maximal normal subgroup of | C2×#13353, #498940, #498941, … |
| maximal quotient of | #142224, #142228, #142273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6341);