Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3483 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
938 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)(3,7)(4,6)(5,8)(9,10,14,12,13,16,22,21)(11,17,20,24,18,15,23,19);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,24)(22,23);(3,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,24)(22,23);(2,6)(5,7)(11,18)(15,17)(19,24)(20,23);(9,12,14,16,13,21,22,10)(11,19,20,17,18,24,23,15)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=dfh,e^a=eh,c^b=cfh,d^b=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129596, #129658, #210196, … |
| maximal normal subgroup of | #129596, #129658, #210196, … |
| maximal quotient of | #6370408, #6370416, #6370468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53308);