Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,13,15)(11,12,14,16)(17,18,21,19)(20,23,24,22);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,14)(15,16)(18,19)(22,23);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,21,18)(20,23,24,22);(17,20)(18,22)(19,23)(21,24);(1,4,5,3)(2,7,8,6)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #97796, #252335, #252336, … |
| maximal normal subgroup of | #97796, #252335, #252336, … |
| maximal quotient of | #521348, #522083, #522705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30946);