Finite Group Databasebeta

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Cayley diagram

C2×#23998 #6451729

order512
nilpotency class3
isoclinismisoclinic class of #23998
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23998

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

4246 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)(5,11)(6,15)(7,10)(8,14)(9,17)(12,16)(13,18);(3,4)(5,12,13,8)(6,9)(7,10)(11,14,18,16)(15,17);(3,5)(4,8)(6,16)(7,13)(9,18)(10,12)(11,15)(14,17);(6,17)(9,15)(11,18)(14,16)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6499855, #72219645, #72219654, …
maximal normal subgroup ofC2×#6499855, #72219645, #72219654, …
maximal quotient of#72219654, #72219672, #72233837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358925);