Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
2582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8);(5,6)(7,8)(9,10,14,17)(11,18,21,15)(12,13,16,22)(19,24,23,20);(1,2)(3,4)(5,6)(7,8)(9,11,14,21)(10,15,17,18)(12,19,16,23)(13,20,22,24);(9,12)(10,13)(11,19)(14,16)(15,20)(17,22)(18,24)(21,23);(9,13)(10,16)(11,20)(12,17)(14,22)(15,23)(18,19)(21,24)Polycyclic presentation:
2^8|b^2=h,c^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #36078⋊C3, #72268900 |
| maximal subgroup of | #132381, #248650, #248651, … |
| maximal normal subgroup of | #132381, #248650, #248651, … |
| maximal quotient of | #517608, #517609, #517616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29066);