Finite Group Databasebeta

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Cayley diagram

C2×#21329 #6372217

order512
nilpotency class3
isoclinismisoclinic class of #3055
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#21329

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 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3774 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)(5,9)(6,7)(8,10)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(4,7)(8,9)(11,13)(12,17,19,25)(14,22)(15,23)(16,21)(18,20,24,26);(3,5,6,10)(4,8,7,9);(13,21)(17,25)(20,26)(22,23)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303341, #72303398, C2×#233589, …
maximal normal subgroup of#72303341, #72303398, C2×#233589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279413);