Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,12)(13,14,19,18)(15,22,17,20)(16,24,26,21)(23,27,25,28);(2,4)(6,9)(7,11)(13,15,19,17)(14,20,18,22)(16,25,26,23)(21,28,24,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,16,19,26)(14,21,18,24)(15,23,17,25)(20,27,22,28);(13,17,19,15)(14,22,18,20)(16,25,26,23)(21,28,24,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,18,19,14)(15,22,17,20)(16,24,26,21)(23,28,25,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6533699, #6533703, #6533709, … |
| maximal normal subgroup of | #6533699, #6533703, #6533709, … |
| maximal quotient of | #521331, #522073, #522581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30771);