Finite Group Databasebeta

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Cayley diagram

C2×#28134 #6459450

order512
nilpotency class3
isoclinismisoclinic class of #24278
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28134

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3003C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6467260, #72239135, #72239138, …
maximal normal subgroup ofC2×#6467260, #72239135, #72239138, …
maximal quotient of#72239135

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366646);