Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,10,9)(7,13,15,11)(8,16,18,12)(14,19,17,20)(21,22,26,29)(23,27,31,25)(24,28,34,36)(30,35,32,33);(6,9)(11,13)(12,16)(19,20)(21,23,26,31)(22,27,29,25)(24,30,34,32)(28,35,36,33);(1,2)(3,4)(5,7,10,15)(6,11,9,13)(8,17,18,14)(12,20,16,19)(21,23)(22,27)(24,32)(25,29)(26,31)(28,33)(30,34)(35,36);(5,8,10,18)(6,12,9,16)(7,14,15,17)(11,19,13,20)(21,24,26,34)(22,28,29,36)(23,30,31,32)(25,33,27,35);(5,9,10,6)(7,13,15,11)(8,16,18,12)(14,20,17,19)(21,25,26,27)(22,23,29,31)(24,33,34,35)(28,30,36,32)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=f,d^2=fh,e^2=fh,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 | #132388, #248823, #248834, … |
| maximal normal subgroup of | #132388, #248823, #248834, … |
| maximal quotient of | #544429, #545014, #545367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35966);