Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | Aut(G) order 32768 |
| 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 | |
498 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19,22,30)(18,23,26,32)(20,28,21,29)(24,31,25,27);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)Polycyclic presentation:
2^8|b^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249813, #249823, #249843, … |
| maximal normal subgroup of | #249813, #249823, #249843, … |
| maximal quotient of | #168128, #168167, #168183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11715);