Finite Group Databasebeta

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Cayley diagram

C2×#36560 #7630264

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#36560

Subgroups

subgroupG/H
center, socle, prime radicalC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

7474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537460);