Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
972 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)(7,13)(8,15)(9,17)(10,11)(12,18)(14,16)(19,20)(24,32)(27,34)(30,36)(33,35);(6,11)(8,13)(12,17)(16,19)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(1,2,3,4)(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,9,10,18)(6,12,11,17)(7,14,15,20)(8,16,13,19)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^8|b^2=h,c^2=fg,d^2=h,e^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 | C2×#37193, #7634605, #7634608, … |
| maximal normal subgroup of | C2×#37193, #7634605, #7634608, … |
| maximal quotient of | #553776, #553781, #554031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30181);