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 | |
565 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 183 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)(7,11)(8,9)(10,12)(14,16)(15,19)(21,22,26,28)(23,27,25,24);(5,7)(6,10)(8,11)(9,12)(13,14,17,16)(15,20,19,18)(21,23,26,25)(22,24,28,27);(5,7)(6,11)(8,10)(9,12)(13,15,17,19)(14,18,16,20)(21,24)(22,23)(25,28)(26,27);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,18,19,20)(21,25,26,23)(22,27,28,24)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=f,d^2=h,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 | #104719, #104720, #104721, … |
| maximal normal subgroup of | #104719, #104720, #104721, … |
| maximal quotient of | #155847, #155868, #156386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10064);