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 | |
934 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)(7,11,14,9)(8,12,16,20)(15,19,17,18)(22,23)(25,33)(27,29)(31,36);(7,14)(8,16)(9,11)(12,20)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(1,2,3,4)(5,7)(6,11)(8,17)(9,13)(10,14)(12,18)(15,16)(19,20)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,8,10,16)(6,12,13,20)(7,15,14,17)(9,18,11,19)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,9,10,11)(6,7,13,14)(8,18,16,19)(12,15,20,17)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,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 | C2×#52065, #7708136, #7708142, … |
| maximal normal subgroup of | C2×#52065, #7708136, #7708142, … |
| maximal quotient of | #560452, #560462, #561548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45053);