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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
974 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)(23,27,30,24)(25,28,33,35)(31,34,36,32);(6,7)(9,17)(11,13)(15,20)(23,30)(24,27)(25,33)(28,35);(1,2)(3,4)(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(21,23,26,30)(22,27,29,24)(25,31,33,36)(28,34,35,32);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20)(21,24,26,27)(22,23,29,30)(25,32,33,34)(28,31,35,36);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=fh,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×#51414, #7707049, #7707057, … |
| maximal normal subgroup of | C2×#51414, #7707049, #7707057, … |
| maximal quotient of | #560370, #560386, #561467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44402);