Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,13,15)(11,12,14,16)(19,20)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18,21,24)(19,23,20,22);(1,3,4,7)(2,6,5,8)(17,19,21,20)(18,22,24,23);(9,12)(10,14)(11,15)(13,16)(17,20,21,19)(18,23,24,22);(1,2)(3,6)(4,5)(7,8)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fg,d^2=g,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 | C2×#59327, #7666571, #7666665, … |
| maximal normal subgroup of | C2×#59327, #7666571, #7666665, … |
| maximal quotient of | #531693, #531726, #531773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52315);