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 |
| prime radical | #3558 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 465 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:
(1,2,5,7)(3,9,10,16)(4,12,14,8)(6,13,15,11)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(20,29)(24,28)(25,32)(27,30);(4,14)(8,12)(9,16)(11,13)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(2,7)(3,10)(8,12)(11,13)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=g,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 | #250458, #250459, #250466, … |
| maximal normal subgroup of | #250458, #250459, #250466, … |
| maximal quotient of | #168910, #168914, #168920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12075);