Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10,12,13)(11,15,16,14)(18,19)(20,24);(10,13)(11,16)(17,18,21,19)(20,23,24,22);(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=efg,b^2=eg,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #139458, #139485, #139932, … |
| maximal normal subgroup of | #139458, #139485, #139932, … |
| maximal quotient of | #154728, #154808, #154827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8657);