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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 416 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)(13,14,18,22)(15,19,20,28)(16,25,26,24)(17,23,27,21);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,23)(17,24)(18,20)(21,26)(22,28)(25,27);(2,4)(13,16,17,19,18,26,27,28)(14,20,21,25,22,15,23,24);(5,7,8,12)(6,10,9,11)(15,20)(16,26)(19,28)(24,25)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,d^2=e,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 | C2×#28628, #6462983, #6462991, … |
| maximal normal subgroup of | C2×#28628, #6462983, #6462991, … |
| maximal quotient of | #324490, #324494, #324498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21616);