Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,29)(28,35)(30,38)(31,33)(32,39)(34,40)(36,37);(1,2,5,4)(3,8,7,6)(10,12)(11,18)(16,20)(19,24)(25,27,31,28)(26,29,33,35)(30,34,39,36)(32,37,38,40);(3,7)(6,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28,31,27)(26,29,33,35)(30,36,39,34)(32,37,38,40);(1,3,5,7)(2,6,4,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,38)(39,40);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,39)(26,32,33,38)(27,34,28,36)(29,37,35,40)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7228602, #7228606, #7228617, … |
| maximal normal subgroup of | #7228602, #7228606, #7228617, … |
| maximal quotient of | #582270, #582457, #582458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46024);