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 | #2555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,12)(11,16,13,15)(17,18,22,21)(19,20,24,23);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,12,16)(19,24)(20,23);(9,10,14,12)(11,15,13,16)(17,19,22,24)(18,23,21,20);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,16,13,15)(17,20)(18,19)(21,24)(22,23);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,16,12,15)(17,21,22,18)(19,20,24,23)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fh,d^2=f,e^2=fg,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 | #130399, #486026, #486536, … |
| maximal normal subgroup of | #130399, #486026, #486536, … |
| maximal quotient of | #532088, #532302, #532864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52732);