Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 262144 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(7,11)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,27)(26,28);(5,6,9,8)(7,10,11,12)(14,15)(19,21)(20,22)(26,27);(1,2)(3,4)(7,11)(10,12)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(5,7)(6,10)(8,12)(9,11)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,8,9,6)(7,12,11,10)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)Polycyclic presentation:
2^8|b^2=fh,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #111990, #112364, #200610, … |
| maximal normal subgroup of | #111990, #112364, #200610, … |
| maximal quotient of | #556361, #556374, #556386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30621);