Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,17,20,32)(8,13)(9,14)(10,25)(11,16)(12,19,28,33)(15,26)(18,29,21,35)(22,36,34,27)(23,30)(24,31);(2,4)(5,7,10,21,9,19,23,34,11,20,26,18,24,33,8,22)(6,12,15,27,14,17,13,29,16,28,25,36,31,32,30,35);(5,8,9,10,11,23,24,26)(6,13,14,25,16,30,31,15)(7,18,33,34,20,21,19,22)(12,27,32,35,28,36,17,29)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=h,f^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124728, #124787, #124788, … |
| maximal normal subgroup of | #124728, #124787, #124788, … |
| maximal quotient of | #129053, #129068, #129078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5645);