Finite Group Databasebeta

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Cayley diagram

C2×#7854 #180859

order512
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphism?
homologyAbelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C42

Aliases

direct productC2×#7854

Subgroups

subgroupG/H
centerC2×C42C42
commutatorC4C22×C4×C8
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

2054 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);(3,4,6,9,7,10,5,8)(11,12)(13,19,21,25)(14,17)(15,18)(16,20,23,26)(22,24);(11,13)(12,16,18,23)(14,20)(15,21)(17,19,24,25)(22,26);(3,5,7,6)(4,8,10,9)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303344, #72303383, C2×#233589
maximal normal subgroup of#72303344, #72303383, C2×#233589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88055);