Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2404 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 4 generators on 32 points:
(1,2,5,7)(3,8,13,20)(4,14,15,9)(6,18,17,10)(11,21,19,23)(12,22,26,27)(16,24,29,28)(25,31,30,32);(2,7)(3,11,13,19)(8,21,20,23)(9,14)(10,18)(12,25,26,30)(22,31,27,32)(24,28);(1,3)(2,8)(4,12)(5,13)(6,19)(7,20)(9,22)(10,21)(11,17)(14,27)(15,26)(16,30)(18,23)(24,31)(25,29)(28,32);(1,4)(2,9)(3,12)(5,15)(6,16)(7,14)(8,22)(10,24)(11,25)(13,26)(17,29)(18,28)(19,30)(20,27)(21,31)(23,32)Polycyclic presentation:
2^8|a^2=h,b^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,e^c=eh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #508106, #508108, #508112, … |
| maximal normal subgroup of | #508106, #508108, #508112, … |
| maximal quotient of | #251677, #251678, #251679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16981);