Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,26)(7,18,9,13,20,15,25,29)(8,21,23,32,24,31,27,14)(19,34,35,30,36,28,22,33);(1,2)(3,4)(5,7)(6,13)(8,19)(9,11)(10,25)(12,20)(14,28)(15,16)(17,29)(18,26)(21,33)(22,27)(23,35)(24,36)(30,31)(32,34);(2,4)(5,8,12,24)(6,14,17,32)(7,19,20,36)(9,22,25,35)(10,23,11,27)(13,28,29,34)(15,30,18,33)(16,31,26,21);(5,9)(6,15)(7,10)(8,22)(11,20)(12,25)(13,26)(14,30)(16,29)(17,18)(19,23)(21,28)(24,35)(27,36)(31,34)(32,33)Polycyclic presentation:
2^8|a^2=f,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^c=fh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24470, #6437930, #6437932, … |
| maximal normal subgroup of | C2×#24470, #6437930, #6437932, … |
| maximal quotient of | #268215, #268217, #268220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17458);