Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,8,7,4)(10,14)(12,16)(17,18,21,20)(19,22,23,24);(3,7)(4,8)(9,10,13,14)(11,12,15,16)(17,19)(18,22)(20,24)(21,23);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,12,14,16);(1,4,6,8)(2,7,5,3)(17,20,21,18)(19,22,23,24);(1,5,6,2)(3,8,7,4)(9,12)(10,15)(11,14)(13,16)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=h,e^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 | #98756, #171865, #171866, … |
| maximal normal subgroup of | #98756, #171865, #171866, … |
| maximal quotient of | #531832, #531924, #532344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52867);