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×#548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1461 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)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22);(2,4)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24);(1,2)(3,4)(5,6,7,8);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(1,2)(3,4)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=f,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 | #497371, #497375, #6533551, … |
| maximal normal subgroup of | #497371, #497375, #6533551, … |
| maximal quotient of | #521355, #522089, #522601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30663);