Finite Group Databasebeta

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Cayley diagram

C2×#20658 #6370806

order512
nilpotency class3
isoclinismisoclinic class of #465
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#20658

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6379134, #72264835, #72264836, …
maximal normal subgroup ofC2×#6379134, #72264835, #72264836, …
maximal quotient ofC2×#188806, #72318107

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278002);