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 | #2631 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
989 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)(7,13,14,11)(9,12,17,16)(15,19,20,18)(22,26)(23,29)(28,33)(30,35);(7,14)(9,17)(11,13)(12,16)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2,3,4)(5,7,10,14)(6,11,8,13)(9,15,17,20)(12,18,16,19)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,8,10,6)(7,11,14,13)(9,16,17,12)(15,18,20,19)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fg,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 | #482887, #6353248, #6353258, … |
| maximal normal subgroup of | #482887, #6353248, #6353258, … |
| maximal quotient of | #544553, #545099, #545549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35163);