Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| 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 | |
529 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(2,4)(5,6,8,9)(7,10,12,11)(14,15)(16,20)(21,22,25,28)(23,27,24,26);(6,9)(10,11)(13,14,17,15)(16,18,20,19)(21,23,25,24)(22,26,28,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,14)(16,18,20,19)(21,22)(23,27)(24,26)(25,28);(5,7,8,12)(6,10,9,11)(13,16,17,20)(14,18,15,19)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=eg,d^2=e,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 | #104194, #104210, #104226, … |
| maximal normal subgroup of | #104194, #104210, #104226, … |
| maximal quotient of | #155739, #155822, #155867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9951);