Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,9)(5,21)(6,10)(7,15)(8,19)(11,22)(13,30)(14,29)(16,24)(17,25)(18,26)(20,27)(23,32)(28,31);(1,3,6,17)(2,7,10,24)(4,13,20,32)(5,16,22,15)(8,23,26,30)(9,28,27,29)(11,25,21,12)(14,19,31,18);(1,4)(2,8)(3,13)(5,18)(6,20)(7,23)(9,21)(10,26)(11,27)(12,28)(14,16)(15,31)(17,32)(19,22)(24,30)(25,29);(1,4)(2,9)(3,14)(5,19)(6,20)(7,23)(8,21)(10,27)(11,26)(12,29)(13,16)(15,32)(17,31)(18,22)(24,30)(25,28);(2,10)(7,24)(8,26)(9,27)(11,21)(12,25)(23,30)(28,29)Polycyclic presentation:
2^8|b^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6573700, #6573701, #6573703, … |
| maximal normal subgroup of | #6573700, #6573701, #6573703, … |
| maximal quotient of | #6358427, #6358429, #6358432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54095);