Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,9)(8,11)(10,12)(13,14,18,16)(15,20,19,17);(2,4)(6,7)(10,11)(13,15)(14,17)(16,20)(18,19);(5,6,9,7)(8,10,12,11)(13,14,18,16)(15,17,19,20);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,20,19,17);(5,8,9,12)(6,10,7,11)(13,17,18,20)(14,19,16,15)Polycyclic presentation:
2^8|a^2=fg,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6994453, #6994457, #6994517, … |
| maximal normal subgroup of | #6994453, #6994457, #6994517, … |
| maximal quotient of | #531613, #531629, #531667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52014);