Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 276 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)(3,14)(4,8)(5,13)(6,11)(7,12)(9,16)(10,18)(15,29)(17,28)(19,24)(20,23)(21,27)(22,30)(25,32)(26,31);(2,8)(4,18)(5,6)(9,25)(10,13)(11,24)(12,23)(14,29)(15,30)(16,21)(19,20)(22,27)(26,28)(31,32);(1,3)(2,9)(4,15)(5,21)(6,16)(7,17)(8,22)(10,26)(11,28)(12,27)(13,14)(18,30)(19,32)(20,31)(23,25)(24,29)Polycyclic presentation:
2^8|d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13310, #498817, #499269, … |
| maximal normal subgroup of | C2×#13310, #498817, #499269, … |
| maximal quotient of | #142242, #142248, #142250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6298);