Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
992 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)(5,6,10,8)(7,11,13,15)(9,12,17,16)(14,18,19,20)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,33,36,35);(7,13)(9,17)(11,15)(12,16)(22,27)(23,30)(29,32)(31,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,23)(22,28)(24,31)(25,27)(26,30)(29,35)(32,33)(34,36);(5,8,10,6)(7,11,13,15)(9,16,17,12)(14,18,19,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,33,28,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,31,32,36)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482905, #6353265, #6353275, … |
| maximal normal subgroup of | #482905, #6353265, #6353275, … |
| maximal quotient of | #544541, #545095, #545541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35156);