Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
914 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)(3,7)(4,5)(6,8)(10,13)(11,15)(18,19)(23,24);(2,5)(6,7)(9,10,14,13)(11,16,15,12)(17,18,22,19)(20,23,21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,22,18)(20,24,21,23);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,15,13,11)(17,20,22,21)(18,23,19,24);(1,3,4,8)(2,6,5,7)(9,13,14,10)(11,16,15,12)(17,21,22,20)(18,24,19,23)Polycyclic presentation:
2^8|b^2=g,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #161600, #161884, #162380, … |
| maximal normal subgroup of | #161600, #161884, #162380, … |
| maximal quotient of | #516512, #516539, #516686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28102);