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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24);(5,6,8,7)(11,18)(15,22)(19,23)(20,24);(1,2)(3,4)(5,7,8,6)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24);(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129783, #214709, #214724, … |
| maximal normal subgroup of | #129783, #214709, #214724, … |
| maximal quotient of | #517708, #517709, #517742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29008);