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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
992 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)(6,10)(11,14)(12,17)(18,20)(21,22)(23,25)(24,28)(26,34)(27,30)(29,35)(31,36)(32,33);(6,10)(7,13)(8,16)(18,20)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(1,2)(3,4)(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,25,27,28)(22,23,30,24)(26,33,35,36)(29,31,34,32);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)Polycyclic presentation:
2^8|a^2=gh,b^2=f,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 | #132174, #247004, #247032, … |
| maximal normal subgroup of | #132174, #247004, #247032, … |
| maximal quotient of | #560530, #561624, #563078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44742);