Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
846 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)(11,12)(13,15)(14,16)(17,18,21,20)(19,23,24,22);(2,3)(6,7)(9,11,13,16)(10,12,15,14)(17,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,15)(13,16)(17,18,21,20)(19,22,24,23);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22);(1,4)(2,6)(3,7)(5,8)(9,12,13,14)(10,11,15,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7091174, #7091188, #7091237, … |
| maximal normal subgroup of | #7091174, #7091188, #7091237, … |
| maximal quotient of | #531998, #532112, #532208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52551);