Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | Aut(G) = #72317013 (order 4096) · Inn(G) = #2021 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
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 | |
369 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 369 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)(4,9)(5,8)(6,11)(7,13)(10,18)(14,25)(15,27)(16,22)(17,24)(19,32)(20,28)(21,29)(23,31)(26,30);(1,3)(2,7)(4,13)(5,14)(6,15)(8,21)(9,12)(10,19)(11,20)(16,29)(17,28)(18,30)(22,25)(23,26)(24,27)(31,32);(2,8)(5,18)(7,19)(9,22)(10,11)(12,25)(13,28)(14,30)(16,31)(20,21)(23,24)(26,27)Polycyclic presentation:
2^8|e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,g^b=gh,d^c=dg,e^c=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151150, #151151, #151322, … |
| maximal normal subgroup of | #151150, #151151, #151322, … |
| maximal quotient of | #142224, #142272, #142320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6338);