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C5×#26857 #11949577

order1280
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C23

Aliases

direct productC5×#26857

Subgroups

subgroupG/H
center, socle, prime radicalC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 1280

A minimal faithful permutation representation — 4 generators on 29 points:

(1,2,3,4,5)(6,7,10,12)(8,13,9,11)(14,15)(16,19)(17,18)(20,21)(23,26)(27,29);(6,8,10,9)(7,11,12,13)(23,27)(25,28)(26,29);(6,9,10,8)(7,11,12,13)(15,18)(19,21)(22,23)(24,26)(25,29)(27,28);(7,12)(11,13)(14,16,17,20)(15,19,18,21)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 19845);