Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2437 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)(7,16)(8,13)(9,14)(10,23)(11,15)(12,18)(17,30)(19,26)(20,28)(21,29)(22,33)(24,27)(25,32)(31,36)(34,35);(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);(5,8)(6,13)(7,17)(9,21)(10,22)(11,20)(12,25)(14,29)(15,28)(16,30)(18,32)(19,31)(23,33)(24,34)(26,36)(27,35);(8,20)(13,28)(17,31)(21,22)(25,35)(29,33)(30,36)(32,34);(5,9,11,10)(6,14,15,23)(7,18,19,24)(8,21,20,22)(12,26,27,16)(13,29,28,33)(17,32,31,34)(25,36,35,30)Polycyclic presentation:
2^8|a^2=g,e^2=h,f^2=h,b^a=bf,f^a=fh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60086⋊C3 |
| maximal subgroup of | #356109, #356110, #359686, … |
| maximal normal subgroup of | #356109, #356110, #359686, … |
| maximal quotient of | #6342907, #6342927, #6342933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53074);