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 | C2×#555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1143 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)(7,11)(10,12)(13,14,18,16)(15,19,21,22)(17,20,25,24)(23,26,27,28);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(15,21)(19,22)(23,27)(26,28);(1,2)(3,4)(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(5,8,9,6)(7,12,11,10)(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=fg,c^2=fg,d^2=fgh,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 | #132375, #248704, #248705, … |
| maximal normal subgroup of | #132375, #248704, #248705, … |
| maximal quotient of | #517829, #517831, #517838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29065);