Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1409 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)(12,17)(13,19)(14,20)(15,18);(5,6,10,12)(7,14,16,19)(8,11,9,17)(13,18,20,15);(2,4)(6,12)(11,17)(13,20)(14,19);(5,7)(6,13)(8,15)(9,18)(10,16)(11,19)(12,20)(14,17);(5,8,10,9)(6,11,12,17)(7,15,16,18)(13,19,20,14)Polycyclic presentation:
2^8|b^2=h,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,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130785, #499750, #500158, … |
| maximal normal subgroup of | #130785, #499750, #500158, … |
| maximal quotient of | #6358591, #6358598, #6358610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54051);