Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1144 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)(6,7)(11,15)(14,16)(17,18)(19,24)(20,23)(21,22);(1,2,5,4)(3,7,8,6)(10,13)(14,16)(18,21)(20,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,20,22,24)(18,23,21,19);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^8|b^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53179⋊C3 |
| maximal subgroup of | C2×#53179, #8105668, #8105743, … |
| maximal normal subgroup of | C2×#53179, #8105668, #8105743, … |
| maximal quotient of | #606719, #606858, #606894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46167);