Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13)(8,17)(9,16)(10,18)(11,15)(12,14)(19,20);(5,7,10,16,11,8,12,19)(6,9,14,13,15,20,18,17);(1,2,3,4)(5,8,11,7)(6,13,15,17)(9,14,20,18)(10,19,12,16);(5,9,10,13,11,20,12,17)(6,7,14,16,15,8,18,19)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25807, #6443285, #6443286, … |
| maximal normal subgroup of | C2×#25807, #6443285, #6443286, … |
| maximal quotient of | #299301, #299304, #299307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18795);