Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | C2×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1325 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)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11)(14,17)(15,19);(1,2,3,4)(13,14,16,17)(15,18,19,20);(5,7,8,12)(6,10,9,11);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|c^2=gh,d^2=f,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 | #480912, #497528, #6533271, … |
| maximal normal subgroup of | #480912, #497528, #6533271, … |
| maximal quotient of | #521699, #522315, #522818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31066);