Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
461 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 461 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,17,20)(15,22,24,19)(16,26,21,18)(23,25,27,28);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,22)(21,27)(26,28);(2,4)(13,16,18,14,17,21,26,20)(15,23,25,19,24,27,28,22);(5,7,8,12)(6,10,9,11)(14,20)(15,24)(16,21)(25,28)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #149992, #149993, #152966, … |
| maximal normal subgroup of | #149992, #149993, #152966, … |
| maximal quotient of | #324492, #324496, #324500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21618);