Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,16,24,29)(11,15,21,27,19,25,32,36)(12,17,22,30,23,28,34,38)(20,26,33,37,31,35,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(3,8)(5,7)(10,16)(11,19)(17,28)(18,29)(20,31)(21,32)(30,38)(33,39);(2,6)(5,7)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(9,12,13,22,14,23,24,34)(10,17,18,30,16,28,29,38)(11,20,21,33,19,31,32,39)(15,26,27,37,25,35,36,40)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cfh,d^b=dh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #60307⋊C3, #72342906 |
| maximal subgroup of | #129593, #129616, #129656, … |
| maximal normal subgroup of | #129593, #129616, #129656, … |
| maximal quotient of | #6370402, #6370456, #6370565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53295);