Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
482 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,32)(29,31);(1,2)(3,8)(4,12)(5,7)(6,9)(10,16)(11,15)(13,14)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,28)(27,30);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,15)(12,16)(20,30)(24,28)(25,32)(27,29);(2,9)(7,12)(8,15)(14,16)(18,23)(19,26)(24,28)(27,29)Polycyclic presentation:
2^8|e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh31 conjugacy classes — view the full character table.
| maximal subgroup of | #250455, #250462, #250463, … |
| maximal normal subgroup of | #250455, #250462, #250463, … |
| maximal quotient of | #168909, #168913, #168919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12074);