Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
353 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15);(2,7)(6,14)(8,12)(9,15)(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(3,10)(6,14)(9,15)(11,16)(19,26)(22,31)(25,32)(27,30)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #232156, #232163, C2×#19752, … |
| maximal normal subgroup of | #232156, #232163, C2×#19752, … |
| maximal quotient of | #191533, #206316, #206317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12740);