Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | C2×#121 | C22 |
| 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) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(8,16)(11,18)(14,20)(17,19)(21,22,26,30)(23,28,24,27)(25,33,34,29)(31,35,32,36);(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19)(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,24,26,23)(22,28,30,27)(25,31,34,32)(29,35,33,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,25,26,34)(22,29,30,33)(23,31,24,32)(27,35,28,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,36)(33,34)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #113075, #113081, #113540, … |
| maximal normal subgroup of | #113075, #113081, #113540, … |
| maximal quotient of | #556364, #556366, #556377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30545);