Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2403 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
933 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 933 conjugacy classes of subgroups — too many to draw. Only its 141 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 — 4 generators on 16 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16);(2,9)(4,5)(7,11)(8,10)(12,14)(15,16)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501188, #501190, #501196, … |
| maximal normal subgroup of | #501188, #501190, #501196, … |
| maximal quotient of | #244583, #244584, #244585, … |
| automorphism group of | C2×D8, C2×Q16 |
| holomorph of | C2×C8 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16870);