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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
946 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)(5,6,10,13)(7,8,11,15)(9,12,17,19)(14,16,18,20)(21,22,26,27)(23,28,31,35)(24,29,30,25)(32,33,34,36);(7,11)(8,15)(9,17)(12,19)(22,27)(23,31)(25,29)(32,34);(1,2,3,4)(5,7)(6,8)(9,14)(10,11)(12,16)(13,15)(17,18)(19,20)(21,23)(22,28)(24,34)(25,33)(26,31)(27,35)(29,36)(30,32);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,24,26,30)(22,29,27,25)(23,32,31,34)(28,33,35,36);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25,26,29)(22,30,27,24)(23,33,31,36)(28,32,35,34)Polycyclic presentation:
2^8|a^2=h,c^2=g,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×#48829, #7704772, #7704955, … |
| maximal normal subgroup of | C2×#48829, #7704772, #7704955, … |
| maximal quotient of | #560419, #560438, #561526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41817);