Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,7)(6,8)(10,13)(11,15)(17,18,21,19)(20,22,24,23);(2,5)(6,8)(10,13)(14,16)(18,19)(20,24)(25,26,28,30)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=ef,d^2=efg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16233, #610482, #610491, … |
| maximal normal subgroup of | C2×#16233, #610482, #610491, … |
| maximal quotient of | #154900, #154919, #154972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9221);