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 | #2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1160 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)(6,10)(11,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(5,6)(7,11)(8,12)(9,10)(13,15,19,17)(14,20,16,22)(18,23,26,25)(21,27,24,28);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,16,19,14)(15,20,17,22)(18,24,26,21)(23,27,25,28);(5,7,9,8)(6,11,10,12)(13,17,19,15)(14,20,16,22)(18,23,26,25)(21,28,24,27);(5,8,9,7)(6,12,10,11)(13,18,19,26)(14,21,16,24)(15,23,17,25)(20,27,22,28)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132382, #248727, #248738, … |
| maximal normal subgroup of | #132382, #248727, #248738, … |
| maximal quotient of | #517609, #517610, #517619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29068);