Finite Group Databasebeta

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Cayley diagram

#41×C22⋊Q8 #7630330

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct product#41×C22⋊Q8

Subgroups

subgroupG/H
centerC22×C4C25
commutator, Frattini, socle, cosocleC23C26
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

8652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#41×#525, #72304543, #72304616, …
maximal normal subgroup of#41×#525, #72304543, #72304616, …
maximal quotient of#72305800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537526);