Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
970 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)(7,12)(8,13)(9,17)(10,11)(14,18)(15,16)(19,20);(6,11)(8,12)(14,17)(16,19)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(1,2)(3,4)(5,7,10,13)(6,12,11,8)(9,15,18,20)(14,19,17,16)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,8,10,12)(6,13,11,7)(9,16,18,19)(14,20,17,15)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,9)(6,14)(7,15)(8,16)(10,18)(11,17)(12,19)(13,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^8|b^2=h,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132095, #246935, #246962, … |
| maximal normal subgroup of | #132095, #246935, #246962, … |
| maximal quotient of | #560366, #560374, #561462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45061);