Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,8,10)(7,9,12,11)(13,14)(15,19)(16,18)(17,20);(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,17,20,19);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20);(13,15,16,20)(14,17,18,19);(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98414, #130559, #130561, … |
| maximal normal subgroup of | #98414, #130559, #130561, … |
| maximal quotient of | #521493, #522145, #522650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32735);