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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,12,9,13,8,18)(7,14,17,20,16,11,15,19);(2,4)(5,7,10,17,9,16,8,15)(6,11,12,19,13,14,18,20);(5,6)(7,14)(8,12)(9,13)(10,18)(11,16)(15,20)(17,19);(2,4)(5,8,9,10)(6,12,13,18)(7,15,16,17)(11,19,14,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=h,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×#25141, #6438879, #6438882, … |
| maximal normal subgroup of | C2×#25141, #6438879, #6438882, … |
| maximal quotient of | #281532, #281533, #281544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18129);