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 |
| prime radical | C2×#547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(10,14)(11,17)(16,19)(18,23);(2,3)(9,10,13,14)(11,15,17,21)(12,19,20,16)(18,24,23,22);(1,2,4,3)(5,6)(7,8);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(1,3,4,2)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)Polycyclic presentation:
2^8|b^2=f,c^2=h,e^2=fh,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 | #130747, #248539, #248540, … |
| maximal normal subgroup of | #130747, #248539, #248540, … |
| maximal quotient of | #521403, #522127, #522678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32325);