Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| 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 | C22×C4 | C24 |
| prime radical | #3558 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
477 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 111 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(20,21)(24,29)(26,31)(27,30);(2,8)(3,10)(6,9)(11,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #244551, #244554, #244556, … |
| maximal normal subgroup of | #244551, #244554, #244556, … |
| maximal quotient of | #168945, #168964, #169001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11454);