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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,14)(11,15,16,12)(18,22)(23,26)(24,29)(30,32);(2,6)(3,7)(10,14)(12,15)(18,22)(19,25)(20,28)(30,32);(9,11)(10,12)(13,16)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3,5,7)(2,4,6,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,4)(2,7)(3,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=gh,d^2=fg,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 | #133512, #500061, #500409, … |
| maximal normal subgroup of | #133512, #500061, #500409, … |
| maximal quotient of | #561210, #562405, #564618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41937);