Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
1333 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)(5,6,10,14)(7,15,17,28)(8,12,20,26)(9,22,16,13)(11,23,24,19)(18,29,31,36)(21,33,30,27)(25,34,35,32);(5,7,16,23,10,17,9,19)(6,11,22,28,14,24,13,15)(8,18,30,34,20,31,21,32)(12,25,33,36,26,35,27,29);(2,4)(7,17)(11,24)(15,28)(18,31)(19,23)(25,35)(29,36)(32,34);(5,8)(6,12)(7,18)(9,21)(10,20)(11,25)(13,27)(14,26)(15,29)(16,30)(17,31)(19,32)(22,33)(23,34)(24,35)(28,36);(8,20)(12,26)(18,31)(21,30)(25,35)(27,33)(29,36)(32,34)Polycyclic presentation:
2^8|a^2=h,b^2=fh,f^2=h,b^a=bf,c^a=cg,f^a=fh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60462⋊C3 |
| maximal subgroup of | #129778, #129780, #481439, … |
| maximal normal subgroup of | #129778, #129780, #481439, … |
| maximal quotient of | #6353845, #6353848, #6353852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53450);