Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| 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 | |
1013 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)(5,6,8,10)(7,9,11,12)(14,19)(17,25)(20,21)(23,24);(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,22)(18,19)(23,28)(24,27)(25,26);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,16)(14,20,19,21)(17,23,25,24)(22,27,26,28);(5,7,8,11)(6,9,10,12)(13,16,18,15)(14,21,19,20)(17,24,25,23)(22,28,26,27);(13,17,18,25)(14,22,19,26)(15,23,16,24)(20,27,21,28)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129919, #208168, #208182, … |
| maximal normal subgroup of | #129919, #208168, #208182, … |
| maximal quotient of | #516450, #516629, #516717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28307);