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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
489 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)(4,14)(8,20)(9,18)(10,15)(12,24)(13,27)(16,29)(19,22)(21,31)(23,30)(25,26)(28,32);(1,2,5,10)(3,8,12,21)(4,15,14,7)(6,9,17,23)(11,22,24,32)(13,28,26,19)(16,30,29,18)(20,25,31,27);(3,12)(4,16)(7,18)(8,21)(11,24)(13,27)(14,29)(15,30)(19,31)(20,28)(22,32)(25,26);(1,3)(2,8)(4,13)(5,12)(6,11)(7,19)(9,22)(10,21)(14,26)(15,28)(16,25)(17,24)(18,20)(23,32)(27,29)(30,31)Polycyclic presentation:
2^8|b^2=g,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×#18416, #6347017, #6347042, … |
| maximal normal subgroup of | C2×#18416, #6347017, #6347042, … |
| maximal quotient of | #169615, #169627, #169680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11404);