Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(8,16)(11,18)(14,20)(17,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98089, #167789, #167790, … |
| maximal normal subgroup of | #98089, #167789, #167790, … |
| maximal quotient of | #553775, #553780, #554032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30179);