Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1999 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15)(5,20)(6,9)(7,12)(8,18)(10,17)(13,27)(14,29)(16,26)(19,30)(21,24)(22,28)(23,31)(25,32);(1,3)(2,7)(4,16)(5,21)(6,13)(8,24)(9,22)(10,26)(11,15)(12,18)(14,17)(19,31)(20,23)(25,29)(27,30)(28,32);(7,22)(11,27)(23,26)(24,29)Polycyclic presentation:
2^8|d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93780, #93781, #151182, … |
| maximal normal subgroup of | #93780, #93781, #151182, … |
| maximal quotient of | #142240, #142241, #142294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6297);