Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,18)(11,15,20,28)(12,17,23,32)(13,16,25,19)(21,27,33,38)(22,29,34,26)(24,30,36,31)(35,39,40,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,29)(17,27)(18,28)(19,26)(23,33)(24,35)(25,34)(30,39)(31,37)(32,38)(36,40);(1,4,5,2)(3,7,8,6)(10,16)(11,20)(13,25)(15,26)(17,30)(18,19)(21,33)(24,36)(27,37)(28,29)(31,32)(38,39);(3,8)(6,7)(9,12)(10,17)(11,21)(13,24)(14,23)(15,27)(16,30)(18,32)(19,31)(20,33)(22,35)(25,36)(26,37)(28,38)(29,39)(34,40);(10,18)(11,20)(12,23)(16,19)(22,34)(24,36)(27,38)(37,39)Polycyclic presentation:
2^8|a^2=fh,c^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130851, #130853, #250541, … |
| maximal normal subgroup of | #130851, #130853, #250541, … |
| maximal quotient of | #6379328, #6379861, #6380090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54280);