Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,11)(8,9)(12,13)(14,19)(15,20)(16,18);(5,6)(7,15)(8,11)(9,19)(10,14)(12,17)(13,16)(18,20);(2,4)(5,7)(6,12)(8,18)(9,16)(10,17)(11,13)(14,15)(19,20);(5,7)(6,13)(8,16)(9,18)(10,17)(11,12)(14,20)(15,19);(5,8,10,9)(6,11,14,19)(7,16,17,18)(12,20,15,13)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,e^c=eh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #220649, #220651, #220672, … |
| maximal normal subgroup of | #220649, #220651, #220672, … |
| maximal quotient of | #6359115, #6359123, #6359133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54075);