Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
521 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(1,2)(3,4)(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,22)(23,27)(24,26)(25,28);(1,2,4,3)(6,9)(10,12)(21,23,26,25)(22,27,24,28);(2,3)(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,22)(23,27,25,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,17,18,19)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=eg,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #142837, #142865, #144937, … |
| maximal normal subgroup of | #142837, #142865, #144937, … |
| maximal quotient of | #155761, #155818, #156011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10001);