Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,6)(3,10,13,8)(4,14,5,9)(11,15,12,16)(17,18)(19,24)(20,23)(21,25)(22,27)(26,31)(28,29)(30,32);(1,3,7,13)(2,8,6,10)(4,11,5,12)(9,15,14,16)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32);(1,4,7,5)(2,9,6,14)(3,11,13,12)(8,15,10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,15)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32);(1,5,7,4)(2,9,6,14)(3,12,13,11)(8,15,10,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32);(1,6,7,2)(3,10,13,8)(4,14,5,9)(11,16,12,15)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #347452, #347472, #347500, … |
| maximal normal subgroup of | #347452, #347472, #347500, … |
| maximal quotient of | #7627415, #7627416, #7627448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56048);