Finite Group Databasebeta

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Cayley diagram

C2×#11537 #480366

order512
nilpotency class4
isoclinismisoclinic class of #11521
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#11537

Subgroups

subgroupG/H
centerC2×C4#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC22#1466
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,6,11,7,12,19,27,8,9,18,22,20,23,30,34)(5,13,14,29,15,21,31,26,16,28,17,33,32,10,24,25);(4,9)(10,21)(12,23)(13,28);(3,5,7,15,8,16,20,32)(4,10,23,28,9,21,12,13)(6,17,19,24,18,14,30,31)(11,26,34,29,22,25,27,33)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#480440, #72240696, #72240698, …
maximal normal subgroup ofC2×#480440, #72240696, #72240698, …
maximal quotient of#72239984, #72240702

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387562);