Finite Group Databasebeta

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Cayley diagram

C2×#21774 #6374179

order512
nilpotency class3
isoclinismisoclinic class of #3192
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#21774

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C22⋊C4
prime radicalC22×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5186 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)(6,8)(7,10)(11,12,15,18,16,19,14,23)(13,20,22,17,21,26,24,25);(3,5)(4,7)(6,10)(8,9)(13,21)(17,25)(20,26)(22,24);(5,10)(7,9)(11,13)(12,17)(14,24)(15,22)(16,21)(18,26)(19,25)(20,23);(4,8)(7,9)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258210, #72258212, #72258213, …
maximal normal subgroup of#72258210, #72258212, #72258213, …
maximal quotient ofC2×#391706, #72318388

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281375);