Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21);(1,3,5,8)(2,6,7,4)(9,11,12,16)(10,13,14,15)(17,19,20,24)(18,21,22,23);(10,14)(13,15)(19,24)(21,23);(2,7)(3,8)(11,16)(13,15)(18,22)(19,24);(1,4,5,6)(2,8,7,3)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #52294⋊C3 |
| maximal subgroup of | #131258, #484921, #485486, … |
| maximal normal subgroup of | #131258, #484921, #485486, … |
| maximal quotient of | #520948, #520957, #521153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45282);