Finite Group Databasebeta

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Cayley diagram

C2×#8094 #181726

order512
nilpotency class3
isoclinismisoclinic class of #1510
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#8094

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1722 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);(3,4,8,9)(5,7,10,6)(11,12,15,19,16,20,14,18)(13,21,22,17,23,25,24,26);(3,5,7,9,8,10,6,4)(11,13)(12,17)(14,24)(15,22)(16,23)(18,21)(19,25)(20,26);(3,6,8,7)(4,5,9,10)(11,14,16,15)(12,18,20,19)(13,22,23,24)(17,25,26,21)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#181727, #72239971, #72240026
maximal normal subgroup ofC2×#181727, #72239971, #72240026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88922);