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 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1431 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)(14,17)(15,19);(6,9)(7,11)(13,14,16,17)(15,18,19,20);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20);(13,15,16,19)(14,18,17,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|b^2=f,c^2=fg,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 | #98434, #130506, #131010, … |
| maximal normal subgroup of | #98434, #130506, #131010, … |
| maximal quotient of | #521557, #522255, #522820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31945);