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 | |
996 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22)(23,25)(24,32)(26,27)(28,31)(29,35)(30,36)(33,34);(6,7)(11,13)(12,14)(18,19)(22,27)(25,28)(29,32)(34,36);(1,2,3,4)(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,23)(22,28)(24,33)(25,27)(26,31)(29,34)(30,35)(32,36);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,24,26,35)(22,29,27,32)(23,30,31,33)(25,34,28,36);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25,26,28)(22,23,27,31)(24,34,35,36)(29,30,32,33)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=f,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 | #482873, #6353203, #6353213, … |
| maximal normal subgroup of | #482873, #6353203, #6353213, … |
| maximal quotient of | #544547, #545096, #545539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34410);