Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,15,16)(19,20)(22,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18,21,24)(19,23,20,22);(17,19,21,20)(18,22,24,23);(9,12,13,10)(11,14,15,16)(17,20,21,19)(18,23,24,22);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=g,d^2=h,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.
| Fitting subgroup of | #59875⋊C3 |
| maximal subgroup of | #97510, #98716, #171871, … |
| maximal normal subgroup of | #97510, #98716, #171871, … |
| maximal quotient of | #531816, #532388, #533790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52863);