Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 131072 |
| 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×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,20)(17,18)(22,23)(25,33)(27,29)(31,36);(6,9)(7,13)(12,17)(14,16)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(1,2)(3,4)(5,7,10,13)(6,11,9,15)(8,16,18,14)(12,20,17,19)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,8,10,18)(6,12,9,17)(7,14,13,16)(11,19,15,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,9,10,6)(7,15,13,11)(8,17,18,12)(14,20,16,19)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,c^2=f,d^2=f,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 | #132173, #247005, #247033, … |
| maximal normal subgroup of | #132173, #247005, #247033, … |
| maximal quotient of | #560523, #561626, #563061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44967);