Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,18,22,20)(19,24,23,21);(2,3)(6,7)(10,11)(14,15)(17,19)(18,21)(20,24)(22,23);(9,10,13,11)(12,14,16,15)(17,20,22,18)(19,24,23,21);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,21,22,24)(18,23,20,19)Polycyclic presentation:
2^8|a^2=fg,c^2=fgh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130363, #133715, #486043, … |
| maximal normal subgroup of | #130363, #133715, #486043, … |
| maximal quotient of | #532121, #532250, #532605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52940);