Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
389 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,26,29)(21,24,27,30)(25,32,28,31);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(10,14)(13,16)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(1,3,4,8)(2,6,5,7)(18,22)(19,26)(24,30)(25,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24984, #6447009, #6447014, … |
| maximal normal subgroup of | C2×#24984, #6447009, #6447014, … |
| maximal quotient of | #280911, #280912, #280913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17972);