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×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1026 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)(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,22,26,28)(23,29,30,25)(24,27,34,36)(31,35,32,33);(6,10)(11,13)(12,17)(18,19)(23,30)(25,29)(31,32)(33,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23)(22,25)(24,32)(26,30)(27,35)(28,29)(31,34)(33,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24)(22,27)(23,31)(25,33)(26,34)(28,36)(29,35)(30,32);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25)(22,23)(24,33)(26,29)(27,31)(28,30)(32,36)(34,35)Polycyclic presentation:
2^8|a^2=h,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 | #215015, #6352447, #6352557, … |
| maximal normal subgroup of | #215015, #6352447, #6352557, … |
| maximal quotient of | #544479, #545054, #545411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34368);