Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| 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 |
| prime radical | #3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 439 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:
(1,2,7,12)(3,13,17,24)(4,18,20,10)(5,8,6,11)(9,14,25,16)(15,27,30,32)(19,22,21,26)(23,28,31,29);(2,8)(3,14)(4,19)(5,6)(10,18)(11,12)(13,24)(16,17)(20,21)(23,27)(28,29)(31,32);(1,3)(2,9)(4,15)(5,14)(6,16)(7,17)(8,13)(10,23)(11,24)(12,25)(18,31)(19,29)(20,30)(21,28)(22,27)(26,32);(1,4)(2,10)(3,15)(5,21)(6,19)(7,20)(8,22)(9,23)(11,26)(12,18)(13,27)(14,28)(16,29)(17,30)(24,32)(25,31)Polycyclic presentation:
2^8|a^2=h,e^2=h,f^2=h,g^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,g^b=gh,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632109, #6632110, #6632111, … |
| maximal normal subgroup of | #6632109, #6632110, #6632111, … |
| maximal quotient of | #331937, #331938, #331939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23430);