Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,24)(22,23);(2,6)(3,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24);(11,16)(14,15)(18,23)(19,20);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,24,23,21);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,14,16,15)(17,21,22,24)(18,19,23,20)Polycyclic presentation:
2^8|a^2=fgh,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133495, #500034, #500382, … |
| maximal normal subgroup of | #133495, #500034, #500382, … |
| maximal quotient of | #561266, #562434, #564610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44803);