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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,12)(13,14)(15,16);(2,6)(3,7)(10,14)(12,15)(18,22)(19,25)(20,28)(30,32);(9,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3)(2,4)(5,7)(6,8)(9,12,13,15)(10,11,14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=fg,d^2=g,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 | #133497, #500038, #500386, … |
| maximal normal subgroup of | #133497, #500038, #500386, … |
| maximal quotient of | #561236, #562420, #564580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44634);