Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(1,2,4,6)(3,5,7,8)(11,17)(13,21)(15,19)(16,20)(26,31)(28,33)(32,37)(36,39);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47780, #7701765, #7701782, … |
| maximal normal subgroup of | C2×#47780, #7701765, #7701782, … |
| maximal quotient of | #576695, #576726, #576767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40768);