Finite Group Databasebeta

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Cayley diagram

C2×#24472 #6437945

order512
nilpotency class3
isoclinismisoclinic class of #24472
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#24472

Subgroups

subgroupG/H
center, socleC22#2410
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4731 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);(3,4,6,11,8,12,7,10)(5,13,16,18,17,14,15,9);(3,5)(4,9)(6,16)(7,15)(8,17)(10,14)(11,13)(12,18);(3,4)(5,14)(6,10)(7,11)(8,12)(9,15)(13,17)(16,18);(3,4)(5,9)(6,11)(7,10)(8,12)(13,16)(14,15)(17,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72215784, #72215961, #72216144, …
maximal normal subgroup of#72215784, #72215961, #72216144, …
maximal quotient of#72226222, #72285483, C4×#24472, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345141);