Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | C2×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
958 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)(6,10)(7,13)(12,17)(14,19)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(6,10)(7,13)(8,15)(18,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^8|b^2=f,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97369, #262201, #262205, … |
| maximal normal subgroup of | #97369, #262201, #262205, … |
| maximal quotient of | #557084, #557385, #557715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39226);