Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
977 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)(8,15)(12,17)(14,19)(18,20)(22,23)(24,31)(25,33)(34,35);(6,10)(7,13)(12,17)(14,19)(21,22,26,23)(24,29,31,27)(25,28,33,30)(32,35,36,34);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,35,36,34);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fh,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 | #132164, #133762, #213991, … |
| maximal normal subgroup of | #132164, #133762, #213991, … |
| maximal quotient of | #553721, #553978, #554197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30108);