Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1019 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)(3,6)(4,5)(7,8)(9,10,13,17,14,18,24,30)(11,15,21,27,19,25,32,36)(12,16,23,29,22,28,34,38)(20,26,33,37,31,35,39,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,31)(23,33)(24,32)(28,35)(29,37)(30,36)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(9,12)(10,16)(11,20)(13,23)(14,22)(15,26)(17,29)(18,28)(19,31)(21,33)(24,34)(25,35)(27,37)(30,38)(32,39)(36,40);(12,22)(16,28)(20,31)(23,34)(26,35)(29,38)(33,39)(37,40)Polycyclic presentation:
2^8|a^2=g,c^2=f,g^2=h,b^a=bf,c^b=ch,e^d=eh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60129⋊C3 |
| maximal subgroup of | #6380994, #6380995, #6380996, … |
| maximal normal subgroup of | #6380994, #6380995, #6380996, … |
| maximal quotient of | #6369783, #6369870, #6369929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53117);