Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14,9,13,16,26)(7,15,19,32,18,11,25,28)(8,20,23,30,22,31,24,12)(17,33,34,27,21,36,35,29);(1,2,4,3)(5,7,16,25,9,18,10,19)(6,11,26,32,13,15,14,28)(8,21,24,34,22,17,23,35)(12,29,31,36,30,27,20,33);(5,8,9,22)(6,12,13,30)(7,17,18,21)(10,24,16,23)(11,27,15,29)(14,31,26,20)(19,35,25,34)(28,33,32,36);(1,3,4,2)(6,13)(11,15)(12,30)(14,26)(20,31)(27,29)(28,32)(33,36)Polycyclic presentation:
2^8|a^2=f,b^2=ef,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25409, #6444319, #6444321, … |
| maximal normal subgroup of | C2×#25409, #6444319, #6444321, … |
| maximal quotient of | #287744, #287750, #287756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18397);