Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1600 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(8,20)(9,18)(10,16)(12,24)(14,28)(17,27)(19,23)(21,31)(22,30)(25,26)(29,32);(1,2,5,10)(3,8,12,21)(4,16,15,7)(6,9,13,22)(11,23,24,32)(14,29,26,19)(17,30,27,18)(20,25,31,28);(3,12)(4,17)(7,18)(8,21)(11,24)(14,28)(15,27)(16,30)(19,31)(20,29)(23,32)(25,26);(1,3,13,24)(2,8,22,32)(4,14,27,28)(5,12,6,11)(7,19,30,31)(9,23,10,21)(15,26,17,25)(16,29,18,20)Polycyclic presentation:
2^8|b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18435, #6347028, #6347045, … |
| maximal normal subgroup of | C2×#18435, #6347028, #6347045, … |
| maximal quotient of | #169638, #169658, #169702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11423);