Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2668 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,16)(9,19)(14,25)(15,24)(17,23)(20,27)(21,26)(22,31)(28,30)(29,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,25)(10,23)(11,13)(12,24)(18,28)(19,29)(20,31)(21,30)(22,26)(27,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,27)(12,26)(13,28)(15,29)(16,30)(17,31)(23,32)(24,25);(3,15)(5,6)(8,23)(10,11)(13,24)(14,29)(16,17)(18,27)(19,21)(22,32)(25,28)(30,31)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632108, #6632110, #6632114, … |
| maximal normal subgroup of | #6632108, #6632110, #6632114, … |
| maximal quotient of | #331999, #332001, #332002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23425);