Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| 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 | |
1313 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)(3,14)(4,10)(5,13)(6,12)(7,11)(8,18)(9,19)(15,23)(16,30)(17,29)(20,28)(21,25)(22,26)(24,31)(27,32);(1,3)(2,8)(4,15)(5,21)(6,18)(7,17)(9,23)(10,27)(11,25)(12,29)(13,14)(16,19)(20,31)(22,32)(24,26)(28,30);(1,4)(2,9)(3,15)(5,19)(6,22)(7,20)(8,23)(10,12)(11,26)(13,28)(14,30)(16,21)(17,31)(18,32)(24,25)(27,29);(1,4)(2,10)(3,16)(5,22)(6,19)(7,20)(8,24)(9,12)(11,28)(13,26)(14,30)(15,21)(17,32)(18,31)(23,25)(27,29);(2,11)(3,17)(9,26)(10,28)(12,13)(15,31)(16,32)(18,21)Polycyclic presentation:
2^8|f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,e^b=eh,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152179, #512880, #512881, … |
| maximal normal subgroup of | #152179, #512880, #512881, … |
| maximal quotient of | #6358842, #6358845, #6358846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54097);