Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | #2679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,15)(13,14)(17,18,21,22)(19,23,24,20);(2,3)(6,7)(10,14)(15,16)(18,22)(19,24);(9,11,13,12)(10,15,14,16)(17,19,21,24)(18,23,22,20);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20,21,23)(18,24,22,19);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,18)(19,23)(20,24)(21,22)Polycyclic presentation:
2^8|a^2=fg,c^2=gh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97878, #172649, #172650, … |
| maximal normal subgroup of | #97878, #172649, #172650, … |
| maximal quotient of | #557189, #557225, #557517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38935);