Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,20,30)(8,13,21,27)(9,15,10,14)(12,24,26,19)(18,31,32,36)(22,29,23,28)(25,34,35,33);(5,7)(6,12)(8,18)(9,19)(10,24)(11,20)(13,25)(14,17)(15,30)(16,26)(21,32)(22,33)(23,34)(27,35)(28,31)(29,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,21)(12,25)(14,28)(15,29)(16,27)(17,31)(19,33)(20,32)(24,34)(26,35)(30,36);(8,21)(13,27)(18,32)(22,23)(25,35)(28,29)(31,36)(33,34);(5,9,11,10)(6,14,16,15)(7,19,20,24)(8,22,21,23)(12,17,26,30)(13,28,27,29)(18,33,32,34)(25,31,35,36)Polycyclic presentation:
2^8|a^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #246324, #246326, #246348, … |
| maximal normal subgroup of | #246324, #246326, #246348, … |
| maximal quotient of | #6346654, #6346663, #6346672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53358);