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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,23)(22,24);(2,3)(6,7)(9,11,12,16)(10,13,14,15)(17,19,21,24)(18,20,23,22);(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,22,24,20);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,23)(21,24);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fh,d^2=gh,e^2=fg,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×#60021, #7665831, #7666091, … |
| maximal normal subgroup of | C2×#60021, #7665831, #7666091, … |
| maximal quotient of | #532177, #532254, #532312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53009);