Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,18)(11,19,20,15)(12,16,22,24)(13,17,23,21)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(11,20)(13,23)(15,19)(17,21)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,16)(14,20)(18,19)(21,22)(23,24)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(9,12)(10,16)(11,21)(13,19)(14,22)(15,23)(17,20)(18,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,23)(10,17,18,21)(11,16,20,24)(12,19,22,15)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41618, #7639714, #7640048, … |
| maximal normal subgroup of | C2×#41618, #7639714, #7640048, … |
| maximal quotient of | #544777, #545227, #545825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34606);