Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(6,9)(7,11)(13,14,16,18)(15,19,20,17)(23,27)(25,28);(1,2,4,3)(6,9)(10,12)(21,22,24,26)(23,25,27,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,17,18,19)(21,23,24,27)(22,25,26,28)Polycyclic presentation:
2^8|a^2=efg,b^2=fh,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #401711, #401917, #401946, … |
| maximal normal subgroup of | #401711, #401917, #401946, … |
| maximal quotient of | #154188, #154388, #154392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8227);