Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 133 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)(3,11,14,7)(4,15)(5,9)(6,10)(8,17)(12,29,30,22)(13,27,20,23)(16,32,31,24)(18,28)(19,26)(21,25);(1,3)(2,7)(4,16)(5,14)(6,20)(8,24)(9,11)(10,27)(12,18)(13,19)(15,32)(17,31)(21,30)(22,25)(23,26)(28,29);(1,4)(2,8)(3,12)(5,17)(6,21)(7,22)(9,15)(10,28)(11,29)(13,31)(14,30)(16,20)(18,19)(23,32)(24,27)(25,26);(1,3)(2,7)(4,12)(5,14)(6,13)(8,22)(9,11)(10,23)(15,29)(16,18)(17,30)(19,20)(21,31)(24,25)(26,27)(28,32)Polycyclic presentation:
2^8|a^2=fh,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501110, #501113, #501114, … |
| maximal normal subgroup of | #501110, #501113, #501114, … |
| maximal quotient of | #267372, #267375, #267382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17397);