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) = C23×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 | |
637 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 637 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(13,14,17,15)(16,18,20,19)(21,22,24,26)(23,25,27,28);(1,2,4,3)(6,9)(10,12)(14,15)(18,19)(23,27)(25,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,25)(24,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=efh,b^2=f,c^2=e,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #102973, #102983, #102993, … |
| maximal normal subgroup of | #102973, #102983, #102993, … |
| maximal quotient of | #154181, #154417, #154430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8197);