Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| 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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
394 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,29)(20,24)(21,28)(22,27)(23,26)(25,31)(30,32);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,20)(18,24,27,29)(21,31,23,30)(25,28,32,26);(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,28,29,26);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,32,29,25)Polycyclic presentation:
2^8|b^2=eg,c^2=gh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29249, #6461197, #6461202, … |
| maximal normal subgroup of | C2×#29249, #6461197, #6461202, … |
| maximal quotient of | #362823, #364146, #368039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22237);