Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2378 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)(5,6,11,15)(7,16,19,25)(8,13,10,23)(9,14,21,29)(12,24,27,17)(18,30,32,35)(20,28,22,33)(26,34,36,31);(5,7)(6,12)(8,17)(9,18)(10,24)(11,19)(13,25)(14,26)(15,27)(16,23)(20,31)(21,32)(22,34)(28,35)(29,36)(30,33);(2,4)(5,8,11,10)(6,13,15,23)(7,17,19,24)(9,20,21,22)(12,25,27,16)(14,28,29,33)(18,31,32,34)(26,35,36,30);(5,9)(6,14)(7,18)(8,20)(10,22)(11,21)(12,26)(13,28)(15,29)(16,30)(17,31)(19,32)(23,33)(24,34)(25,35)(27,36);(9,21)(14,29)(18,32)(20,22)(26,36)(28,33)(30,35)(31,34)Polycyclic presentation:
2^8|a^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60377⋊C3 |
| maximal subgroup of | #6573639, #6573640, #6573641, … |
| maximal normal subgroup of | #6573639, #6573640, #6573641, … |
| maximal quotient of | #6350064, #6350066, #6350068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53365);