Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,29,23)(20,24,22,26)(28,32,30,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(19,28)(20,22)(23,31)(24,26)(27,32)(29,30);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,19,21,29)(18,23,25,27)(20,28,22,30)(24,31,26,32);(3,8)(5,6)(11,15)(13,16)(17,20,21,22)(18,24,25,26)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=eh,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#32729, #6631244, #6631255, … |
| maximal normal subgroup of | C2×#32729, #6631244, #6631255, … |
| maximal quotient of | #391191, #391193, #391232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25717);