Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,7,5)(9,10,13,15)(11,12,14,16)(19,20)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,18,21,24)(19,23,20,22);(1,2,6,4)(3,5,7,8)(17,19,21,20)(18,22,24,23);(1,4,6,2)(3,8,7,5)(9,12)(10,14)(11,15)(13,16)(17,20,21,19)(18,23,24,22);(1,5,6,8)(2,7,4,3)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7035057, #7035058, #7035081, … |
| maximal normal subgroup of | #7035057, #7035058, #7035081, … |
| maximal quotient of | #531701, #531756, #531777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52321);