Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | |
906 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)(9,10,13,15)(11,16,12,14)(17,18,21,24)(19,22,20,23);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,15,16);(2,3)(6,7)(10,15)(14,16)(17,18)(19,22)(20,23)(21,24);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,19,21,20)(18,22,24,23);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,15,14)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=g,b^2=fh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131215, #484892, #485465, … |
| maximal normal subgroup of | #131215, #484892, #485465, … |
| maximal quotient of | #556509, #556512, #556513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30593);