Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1061 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)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,10,13,16)(11,15,17,22)(12,20,19,14)(18,24,23,21);(2,6)(5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24);(11,17)(15,22)(18,23)(21,24)Polycyclic presentation:
2^8|b^2=f,c^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133269, #152107, #152124, … |
| maximal normal subgroup of | #133269, #152107, #152124, … |
| maximal quotient of | #6374312, #6374462, #6374972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53808);