Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 | |
567 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 167 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)(15,17)(16,18)(19,20)(23,27)(25,28);(6,9)(10,12)(14,15)(16,20)(21,22,24,26)(23,25,27,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #125763, #126058, #126126, … |
| maximal normal subgroup of | #125763, #126058, #126126, … |
| maximal quotient of | #154753, #154847, #154934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9069);