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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,23,24,20);(2,4)(5,6,8,7)(10,11)(14,15)(18,22)(19,24);(1,2)(3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,22)(21,24);(9,12)(10,14)(11,15)(13,16)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^8|a^2=ef,b^2=fh,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #295279, #295289, #295304, … |
| maximal normal subgroup of | #295279, #295289, #295304, … |
| maximal quotient of | #154154, #154245, #154256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8000);