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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1405 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)(12,19)(13,20)(14,18)(15,17);(5,6)(7,12)(8,11)(9,18)(10,14)(13,15)(16,20)(17,19);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,19)(13,18)(14,20);(5,7)(6,13)(8,15)(9,17)(10,16)(11,20)(12,18)(14,19);(5,8,10,9)(6,11,14,18)(7,15,16,17)(12,13,20,19)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,f^b=fh,e^c=eh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497339, #497340, #497350, … |
| maximal normal subgroup of | #497339, #497340, #497350, … |
| maximal quotient of | #6359037, #6359042, #6359043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54070);