Finite Group Databasebeta

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Cayley diagram

C2×#21330 #6372218

order512
nilpotency class3
isoclinismisoclinic class of #3057
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#21330

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238815, #72238842
maximal normal subgroup of#72238815, #72238842
maximal quotient ofC2×#358779

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279414);