Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17,19,27)(8,22,25,30)(9,15)(10,16)(11,14)(12,20,28,21)(13,24,29,23)(18,32)(26,33)(31,34)(35,36);(1,2,3,4)(5,7,9,21,11,19,10,20)(6,12,15,27,14,28,16,17)(8,18,24,34,25,33,23,35)(13,26,22,36,29,32,30,31);(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,35)(21,34)(27,36)(28,32);(6,14)(7,19)(13,29)(15,16)(18,33)(20,21)(22,30)(34,35)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137703, #359613, #359614, … |
| maximal normal subgroup of | #137703, #359613, #359614, … |
| maximal quotient of | #193545, #193546, #193547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13907);