Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,16)(10,15)(11,14)(12,19)(13,24)(18,33)(20,28)(21,29)(23,30)(25,31)(26,32)(27,35)(34,36);(2,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,28)(15,29)(16,17)(18,25)(22,33)(24,35)(26,34)(27,32)(31,36);(5,8,11,26)(6,13,14,31)(7,18,21,35)(9,25,10,24)(12,27,28,33)(15,32,16,22)(17,30,29,36)(19,23,20,34);(6,14)(12,28)(13,31)(15,16)(17,29)(22,32)(27,33)(30,36)Polycyclic presentation:
2^8|c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,g^b=gh,e^c=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #376202, #376204, #381484, … |
| maximal normal subgroup of | #376202, #376204, #381484, … |
| maximal quotient of | #232446, #232448, #232450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16428);