Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,9,13,8,12,15,11)(7,14,17,20,16,10,18,19);(1,2,4,3)(5,7,15,18,8,16,9,17)(6,10,11,20,12,14,13,19);(6,11)(7,16)(9,15)(10,19)(12,13)(14,20);(1,3,4,2)(6,12)(10,14)(11,13)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25408, #6444319, #6444320, … |
| maximal normal subgroup of | C2×#25408, #6444319, #6444320, … |
| maximal quotient of | #287743, #287749, #287755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18396);