Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1004 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)(5,6,10,14)(7,8,12,11)(9,13,17,20)(15,16,19,18)(21,22,27,25)(23,30,24,28)(26,33,34,29)(31,35,32,36);(7,12)(8,11)(9,17)(13,20)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,12)(13,18)(16,20)(17,19)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,8,10,11)(6,12,14,7)(9,16,17,18)(13,19,20,15)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,9,10,17)(6,13,14,20)(7,15,12,19)(8,16,11,18)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132180, #246948, #246978, … |
| maximal normal subgroup of | #132180, #246948, #246978, … |
| maximal quotient of | #560363, #560365, #561460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45047);