Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 417 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:
(5,6,9,12)(7,11,8,10)(13,14,18,26,16,19,17,21)(15,22,25,20,23,28,24,27);(1,2,3,4)(5,7,9,8)(6,10,12,11)(14,19)(20,27)(21,26)(22,28);(2,4)(13,15,16,23)(14,20,19,27)(17,25,18,24)(21,28,26,22);(5,8,9,7)(6,11,12,10)(14,19)(20,27)(21,26)(22,28)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #101890, #150006, #153055, … |
| maximal normal subgroup of | #101890, #150006, #153055, … |
| maximal quotient of | #323269, #323277, #323287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21503);