Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | Aut(G) order 4096 |
| 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 | #3515 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18,23,26)(19,27,24,32)(20,28,30,29)(21,25,22,31);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(17,19,23,24)(18,20,26,30)(21,29,22,28)(25,32,31,27);(17,20)(18,24)(19,26)(21,27)(22,32)(23,30)(25,29)(28,31);(2,7)(3,10)(8,12)(11,13)(17,21,23,22)(18,25,26,31)(19,28,24,29)(20,27,30,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18901, #6348865, #6348888, … |
| maximal normal subgroup of | C2×#18901, #6348865, #6348888, … |
| maximal quotient of | #167518, #167533, #167552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11889);