Finite Group Databasebeta

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Cayley diagram

C2×#39372 #7636638

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#39372

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

8382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7697284, C2×#7636648, C2×#7694927
maximal normal subgroup ofC2×#7697284, C2×#7636648, C2×#7694927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7543834);