Finite Group Databasebeta

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Cayley diagram

C2×#30057 #6434127

order512
nilpotency class3
isoclinismisoclinic class of #30056
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#30057

Subgroups

subgroupG/H
center, socleC22#2631
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224753, #72224754, #72224755, …
maximal normal subgroup of#72224753, #72224754, #72224755, …
maximal quotient of#72224754, #72224757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341323);