Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(5,6,9,11)(7,8,12,10)(13,14,17,27,18,22,19,23)(15,24,26,28,16,20,25,21);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15)(14,20)(16,18)(17,26)(19,25)(21,27)(22,24)(23,28);(2,4)(13,16,18,15)(14,21,22,28)(17,26,19,25)(20,23,24,27);(5,8,9,10)(6,7,11,12)(13,17)(14,22)(15,25)(16,26)(18,19)(21,28)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125744, #126087, #126354, … |
| maximal normal subgroup of | #125744, #126087, #126354, … |
| maximal quotient of | #345610, #345624, #345633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24237);