Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(8,12)(10,11)(14,18)(16,19);(1,2)(3,4)(6,7)(10,11)(14,18)(15,20);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18);(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,19,18,16);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^8|c^2=gh,d^2=h,e^2=g,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 | #111670, #112303, #112912, … |
| maximal normal subgroup of | #111670, #112303, #112912, … |
| maximal quotient of | #516487, #516574, #516649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28224);