Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C2×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,13)(14,16)(17,18,20,21)(19,22,24,23);(2,4)(10,13)(11,15)(18,21)(22,23);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #136132, #136212, #137309, … |
| maximal normal subgroup of | #136132, #136212, #137309, … |
| maximal quotient of | #154049, #154056, #154062, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7609);