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) = C23×C4 |
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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,17,16)(15,19,20,18)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(6,9)(10,12)(21,23,24,27)(22,25,26,28);(5,6,8,9)(7,10,11,12)(13,15,17,20)(14,18,16,19)(21,23)(22,25)(24,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=f,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #100932, #100934, #100948, … |
| maximal normal subgroup of | #100932, #100934, #100948, … |
| maximal quotient of | #154152, #154236, #154321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8051);