Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 262144 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(8,15)(12,17)(14,19)(18,20)(21,22,27,30)(23,25,24,28)(26,34,35,29)(31,36,32,33);(6,10)(7,13)(12,17)(14,19)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(1,2)(3,4)(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,25,27,28)(22,23,30,24)(26,33,35,36)(29,31,34,32);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fh,d^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 | #97249, #132092, #132152, … |
| maximal normal subgroup of | #97249, #132092, #132152, … |
| maximal quotient of | #553734, #553995, #554190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29950);