Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 | |
681 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 681 conjugacy classes of subgroups — too many to draw. Only its 199 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)(7,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,28)(23,24,27,26);(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,20,19)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,24)(25,27)(26,28);(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,20)(18,19)(21,24)(22,27)(23,28)(25,26)Polycyclic presentation:
2^8|a^2=egh,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 | #102912, #102922, #102932, … |
| maximal normal subgroup of | #102912, #102922, #102932, … |
| maximal quotient of | #155847, #155850, #156440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10068);