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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
986 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)(13,14,17,20)(15,21,23,18)(16,19,25,28)(22,27,24,26);(6,10)(8,11)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10);(5,7,9,12)(6,11,10,8)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,11)(6,12,10,7)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=h,e^2=fh,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 | #6672013, #6672016, #6672037, … |
| maximal normal subgroup of | #6672013, #6672016, #6672037, … |
| maximal quotient of | #521907, #522466, #522956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32843);