Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1046 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)(6,7)(10,11)(13,14,18,21)(15,16,23,19)(17,25,27,20)(22,28,26,24);(6,7)(8,12)(13,15,18,23)(14,19,21,16)(17,26,27,22)(20,24,25,28);(1,2,3,4)(5,6,9,7)(8,10,12,11)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,28)(26,27);(5,7,9,6)(8,11,12,10)(13,16)(14,15)(17,24)(18,19)(20,22)(21,23)(25,26)(27,28);(5,8)(6,10)(7,11)(9,12)(13,17,18,27)(14,20,21,25)(15,22,23,26)(16,24,19,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97196, #97335, #172175, … |
| maximal normal subgroup of | #97196, #97335, #172175, … |
| maximal quotient of | #557075, #557080, #557385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38986);