Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 524288 |
| 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 | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,9,7)(8,11,12,10)(14,19)(15,16)(17,25)(27,28);(6,7)(10,11)(13,14,18,19)(15,21,16,20)(17,22,25,26)(23,28,24,27);(13,15,18,16)(14,20,19,21)(17,23,25,24)(22,27,26,28);(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,21,19,20)(17,24,25,23)(22,28,26,27);(5,8)(6,10)(7,11)(9,12)(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107221, #107222, #107303, … |
| maximal normal subgroup of | #107221, #107222, #107303, … |
| maximal quotient of | #513784, #513802, #513808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27359);