Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 16384 |
| 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 | #3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1045 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)(9,10,13,17,14,18,24,30)(11,15,21,27,20,26,33,37)(12,16,23,29,22,28,34,38)(19,25,32,36,31,35,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,27)(18,26)(22,31)(23,32)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(3,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,28)(20,31)(21,32)(24,34)(26,35)(27,36)(30,38)(33,39)(37,40);(2,6)(5,7)(11,20)(15,26)(19,31)(21,33)(25,35)(27,37)(32,39)(36,40);(11,20)(12,22)(15,26)(16,28)(21,33)(23,34)(27,37)(29,38)Polycyclic presentation:
2^8|a^2=g,g^2=h,b^a=bf,d^a=df,c^b=cf,d^b=dh,e^b=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129687, #209560, #209561, … |
| maximal normal subgroup of | #129687, #209560, #209561, … |
| maximal quotient of | #6370179, #6370223, #6370307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53226);