Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,4,7,8)(9,10,14,16)(11,13,12,15)(19,23)(22,24);(2,6)(3,7)(9,11,14,12)(10,15,16,13)(17,18,21,20)(19,22,23,24);(10,16)(11,12)(17,19)(18,22)(20,24)(21,23);(1,3)(2,4)(5,7)(6,8)(9,12,14,11)(10,15,16,13)(17,20,21,18)(19,24,23,22);(1,4)(2,3)(5,8)(6,7)(9,13,14,15)(10,11,16,12)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51661, #7707447, #7707666, … |
| maximal normal subgroup of | C2×#51661, #7707447, #7707666, … |
| maximal quotient of | #561224, #561331, #562420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44649);