Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(17,18,21,20)(19,23,24,22);(1,2,5,3)(4,7,8,6)(10,13)(11,15);(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,20,23);(1,3,5,2)(4,7,8,6);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=gh,d^2=f,e^2=g,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 | #6543665, #6543677, #6543687, … |
| maximal normal subgroup of | #6543665, #6543677, #6543687, … |
| maximal quotient of | #521675, #522305, #522793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31250);