Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
1565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24);(2,3)(10,11)(15,17)(16,18)(22,23);(1,2,4,3)(5,6,7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23);(1,3,4,2)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)Polycyclic presentation:
2^8|c^2=fgh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #497308, #497318, #6533259, … |
| maximal normal subgroup of | #497308, #497318, #6533259, … |
| maximal quotient of | #521431, #522148, #522702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32345);