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 | #2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,4)(9,10,14,13)(11,16,15,12)(17,18,21,20)(19,22,23,24);(1,2,6,5)(3,8,4,7)(11,15)(12,16)(17,19)(18,22)(20,24)(21,23);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15);(1,4,6,3)(2,8,5,7)(9,12,14,16)(10,15,13,11)(17,20,21,18)(19,22,23,24);(1,5,6,2)(3,8,4,7)(9,13,14,10)(11,16,15,12)Polycyclic presentation:
2^8|a^2=fh,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 | #98758, #171873, #171874, … |
| maximal normal subgroup of | #98758, #171873, #171874, … |
| maximal quotient of | #531832, #531923, #532343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52868);