Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,8)(7,13,15,11)(9,16,17,12)(14,18,20,19)(21,22,26,28)(23,29,30,24)(25,27,33,34)(31,35,36,32);(6,8)(11,13)(12,16)(18,19)(23,30)(24,29)(25,33)(27,34);(1,2,3,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,23,26,30)(22,24,28,29)(25,31,33,36)(27,32,34,35);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,35)(34,36);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,25)(22,27)(23,31)(24,32)(26,33)(28,34)(29,35)(30,36)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,d^2=f,e^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 | C2×#48834, #7704853, #7704955, … |
| maximal normal subgroup of | C2×#48834, #7704853, #7704955, … |
| maximal quotient of | #560419, #560437, #561523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41822);