Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,12)(7,10,8,11)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27);(5,7,9,8)(6,10,12,11)(14,16)(19,21)(20,24)(26,27);(5,6)(7,10)(8,11)(9,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #6384203, #6384204, #6384219, … |
| maximal normal subgroup of | #6384203, #6384204, #6384219, … |
| maximal quotient of | #513624, #513635, #513652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27295);