Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
471 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)(14,25)(17,26)(19,21)(22,29)(23,31)(30,32);(1,2,5,10)(3,8,11,21)(4,16,15,7)(6,9,12,22)(13,20,24,19)(14,23,25,32)(17,29,26,18)(27,30,28,31);(3,11)(4,17)(7,18)(8,21)(13,27)(14,25)(15,26)(16,29)(19,31)(20,30)(23,32)(24,28);(1,3,12,25)(2,8,22,32)(4,13,26,27)(5,11,6,14)(7,19,29,31)(9,23,10,21)(15,24,17,28)(16,20,18,30)Polycyclic presentation:
2^8|b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18282, #6346582, #6346587, … |
| maximal normal subgroup of | C2×#18282, #6346582, #6346587, … |
| maximal quotient of | #167035, #167047, #167100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11270);