Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | Aut(G) = #72314138 (order 4096) · Inn(G) = #427 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #427 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1835 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15)(4,22,25,53)(5,27)(6,11)(7,30)(8,14)(9,18)(10,24,36,56)(12,16)(13,28)(17,45,48,31)(19,40)(20,32)(21,42)(23,54,44,37)(26,57,55,35)(29,38)(33,51)(34,43)(39,47,60,64)(41,50)(46,52,62,61)(49,58,63,59);(1,3,16,43,28,50,6,19,8,21,5,18,7,20,29,51)(2,9,11,15,30,40,12,32,14,34,38,42,13,33,27,41)(4,17,44,62,55,63,24,47,25,48,23,46,26,49,56,64)(10,31,35,39,54,59,22,52,36,45,57,60,37,58,53,61);(1,4,5,23,28,55,29,56,8,25,16,44,7,26,6,24)(2,10,11,35,30,54,12,22,14,36,38,57,13,37,27,53)(3,17,18,46,50,63,51,64,21,48,43,62,20,49,19,47)(9,31,15,39,40,59,32,52,34,45,42,60,33,58,41,61)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=d,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94210, #94211, #150797, … |
| maximal normal subgroup of | #94210, #94211, #150797, … |
| maximal quotient of | #135662, #135663, #135679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5881);