Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15)(8,13)(9,21)(10,12)(11,17)(14,22)(16,28)(18,27)(19,25)(20,33)(23,31)(24,30)(26,32)(29,35)(34,36);(5,7,9,17,10,18,22,31)(6,11,14,15,12,23,21,27)(8,16,20,30,19,29,32,36)(13,24,26,28,25,34,33,35);(1,2,3,4)(6,12)(11,23)(13,25)(14,21)(15,27)(24,34)(26,33)(28,35);(5,8)(6,13)(7,16)(9,20)(10,19)(11,24)(12,25)(14,26)(15,28)(17,30)(18,29)(21,33)(22,32)(23,34)(27,35)(31,36);(8,19)(13,25)(16,29)(20,32)(24,34)(26,33)(28,35)(30,36)Polycyclic presentation:
2^8|b^2=f,c^2=gh,f^2=h,b^a=bf,c^a=cg,f^a=fh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60649⋊C3 |
| maximal subgroup of | #130005, #483042, #483112, … |
| maximal normal subgroup of | #130005, #483042, #483112, … |
| maximal quotient of | #6357190, #6357191, #6357192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53637);