Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,7,8)(3,4,6,5)(10,11)(15,16)(18,22)(23,24);(1,3,7,6)(2,5,8,4)(9,10,14,11)(12,15,13,16)(17,18,21,22)(19,23,20,24);(1,4,7,5)(2,3,8,6)(9,11,14,10)(12,16,13,15)(17,19,21,20)(18,23,22,24);(1,5,7,4)(2,3,8,6)(9,12,14,13)(10,15,11,16);(1,6,7,3)(2,5,8,4)(9,13,14,12)(10,16,11,15)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=g,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 | C2×#36025, #7629519, #7629540, … |
| maximal normal subgroup of | C2×#36025, #7629519, #7629540, … |
| maximal quotient of | #517710, #517712, #517713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29013);