Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | #2619 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1165 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)(5,6,10,9)(7,12,8,11)(13,14,18,16)(15,19,21,22)(17,20,25,24)(23,26,27,28);(5,7,10,8)(6,11,9,12)(15,21)(19,22)(23,27)(26,28);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(5,9,10,6)(7,12,8,11)(13,16,18,14)(15,22,21,19)(17,20,25,24)(23,26,27,28);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129901, #214815, #214831, … |
| maximal normal subgroup of | #129901, #214815, #214831, … |
| maximal quotient of | #517721, #517723, #517746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29029);