Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(7,11)(10,12)(13,14,17,19)(15,16,20,18);(6,9)(10,12)(13,15,17,20)(14,18,19,16);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97097, #122045, #122187, … |
| maximal normal subgroup of | #97097, #122045, #122187, … |
| maximal quotient of | #516395, #516593, #516619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27970);