Finite Group Databasebeta

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Cayley diagram

C22×#3387 #10576191

order512
nilpotency class3
isoclinismisoclinic class of #3264
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC22×#3387

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

6518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2);(3,4);(5,6,9,8)(7,11,12,10)(13,14,17,20)(15,21,22,28)(16,24,26,19)(18,25,27,23);(5,7,9,12)(6,10,8,11)(15,22)(18,27)(21,28)(23,25);(13,15,17,22)(14,18,20,27)(16,25,26,23)(19,28,24,21);(5,8,9,6)(7,10,12,11)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#20323
maximal normal subgroup ofC22×#20323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10483387);