Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,18,23)(15,20,26,19)(16,21,22,17)(24,25,27,28);(5,6)(7,11)(8,9)(10,12)(13,15,18,26)(14,19,23,20)(16,27,22,24)(17,28,21,25);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,27)(17,25)(18,26)(19,23)(21,28)(22,24);(5,6)(7,10)(8,9)(11,12)(13,16,18,22)(14,21,23,17)(15,24,26,27)(19,28,20,25);(5,7)(6,10)(8,12)(9,11)(13,17,18,21)(14,22,23,16)(15,25,26,28)(19,27,20,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133389, #246348, #246356, … |
| maximal normal subgroup of | #133389, #246348, #246356, … |
| maximal quotient of | #576602, #576661, #576716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39414);