Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 133 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)(15,22,18,19)(16,24,25,28)(20,26,27,23);(1,2)(3,4)(6,9)(10,12)(13,15,17,18)(14,19,21,22)(16,23,25,26)(20,24,27,28);(2,4)(7,11)(10,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,24)(21,27)(22,28);(5,7,8,11)(6,10,9,12)(16,25)(20,27)(23,26)(24,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=de,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #166738, #166744, #166770, … |
| maximal normal subgroup of | #166738, #166744, #166770, … |
| maximal quotient of | #328187, #328196, #328205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22656);