Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,11)(7,13,15,9)(8,16,19,12)(14,18,17,20)(21,22,26,30)(23,31,32,27)(24,28,34,36)(25,29,35,33);(6,11)(9,13)(12,16)(18,20)(23,32)(25,35)(27,31)(29,33);(1,2,3,4)(5,7)(6,9)(8,17)(10,15)(11,13)(12,20)(14,19)(16,18)(21,23,26,32)(22,27,30,31)(24,33,34,29)(25,28,35,36);(5,8,10,19)(6,12,11,16)(7,14,15,17)(9,18,13,20)(21,24)(22,28)(23,29)(25,27)(26,34)(30,36)(31,35)(32,33);(5,9)(6,7)(8,18)(10,13)(11,15)(12,14)(16,17)(19,20)(21,25)(22,29)(23,28)(24,27)(26,35)(30,33)(31,34)(32,36)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482900, #6353261, #6353271, … |
| maximal normal subgroup of | #482900, #6353261, #6353271, … |
| maximal quotient of | #544523, #545085, #545499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34399);