Finite Group Databasebeta

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Cayley diagram

C2×#23990 #6451778

order512
nilpotency class3
isoclinismisoclinic class of #23990
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23990

Subgroups

subgroupG/H
center, socleC22#2404
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4216 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);(6,13)(8,14)(9,16)(12,17);(4,8,10,14)(9,15,16,18);(3,4)(5,9)(6,14)(7,10)(8,13)(11,16)(12,18)(15,17);(3,5)(4,9)(6,12)(7,11)(8,15)(10,16)(13,17)(14,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6499855, #72219665, #72219666, …
maximal normal subgroup ofC2×#6499855, #72219665, #72219666, …
maximal quotient of#72219665, #72233129, #72237004, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358974);