Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| 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 | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 127 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)(7,10)(8,9)(11,12)(13,14,17,21,16,20,25,19)(15,22,24,27,23,18,26,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,23)(17,24)(19,27)(20,22)(21,28)(25,26);(2,4)(14,19)(17,25)(18,27)(20,21)(22,28)(24,26);(5,7,8,11)(6,10,9,12)(15,23)(18,22)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #485013, #485014, C2×#23703, … |
| maximal normal subgroup of | #485013, #485014, C2×#23703, … |
| maximal quotient of | #243361, #243365, #243369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16691);