Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,7)(9,10,13,14)(11,15,16,12)(19,22)(20,24);(1,2,4,3)(5,6,8,7)(10,14)(11,16)(17,18,21,23)(19,20,22,24);(5,6,8,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,22)(18,20,23,24);(1,3,4,2)(5,7,8,6)(9,12,13,15)(10,11,14,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #293316, #293317, #293324, … |
| maximal normal subgroup of | #293316, #293317, #293324, … |
| maximal quotient of | #154136, #154241, #154277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8008);