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 |
| prime radical | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1002 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)(5,6,10,7)(8,11,16,13)(9,14,17,12)(15,19,20,18)(21,22,26,29)(23,24,27,31)(25,28,33,35)(30,32,34,36);(6,7)(9,17)(11,13)(15,20)(23,27)(24,31)(25,33)(28,35);(1,2)(3,4)(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(21,23)(22,24)(25,30)(26,27)(28,32)(29,31)(33,34)(35,36);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=f,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 | #132427, #245632, #245706, … |
| maximal normal subgroup of | #132427, #245632, #245706, … |
| maximal quotient of | #560334, #560395, #561437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41768);