Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| 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 | |
836 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)(3,7)(10,13)(11,15)(17,18,21,19)(20,22,24,23);(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(20,24)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=g,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.
| maximal subgroup of | #98689, #258912, #258914, … |
| maximal normal subgroup of | #98689, #258912, #258914, … |
| maximal quotient of | #531817, #531856, #531912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52627);