Finite Group Databasebeta

← Prev (#243825) Next (#243827) →

Cayley diagram

C2×#10661 #243826

order512
nilpotency class3
isoclinismisoclinic class of #10012
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#10661

Subgroups

subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC2×C42C24
socleC23#399
prime radicalC23×D4Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#244122, C2×#244118, C2×#243831
maximal normal subgroup ofC2×#244122, C2×#244118, C2×#243831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151022);