Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | |
1013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,16)(11,15,12,13)(17,18,21,22)(19,20,23,24);(2,6)(3,7)(11,12)(13,15)(18,22)(19,23);(9,11,14,12)(10,13,16,15)(17,19)(18,20)(21,23)(22,24);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,15,16,13)(17,20)(18,19)(21,24)(22,23);(1,4)(2,7)(3,6)(5,8)(9,13,14,15)(10,11,16,12)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97175, #130081, #133456, … |
| maximal normal subgroup of | #97175, #130081, #133456, … |
| maximal quotient of | #557222, #557546, #557667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39187);