Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | Aut(G) = C2×#72286737 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
485 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 485 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,16)(8,21)(9,19)(10,17)(12,15)(14,29)(18,28)(20,25)(22,32)(23,31)(24,30)(26,27);(1,2,5,10)(3,8,12,22)(4,17,16,7)(6,9,13,23)(11,25,15,24)(14,30,27,20)(18,31,28,19)(21,26,32,29);(3,12)(4,18)(7,19)(8,22)(11,15)(14,26)(16,28)(17,31)(20,21)(24,25)(27,29)(30,32);(1,3,13,11)(2,8,23,25)(4,14,28,26)(5,12,6,15)(7,20,31,21)(9,24,10,22)(16,27,18,29)(17,30,19,32)Polycyclic presentation:
2^8|b^2=g,d^2=gh,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 | #347035, #347036, #347037, … |
| maximal normal subgroup of | #347035, #347036, #347037, … |
| maximal quotient of | #168935, #168951, #168992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11452);