Finite Group Databasebeta

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Cayley diagram

C2×#20008 #6358401

order512
nilpotency class3
isoclinismisoclinic class of D82
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#20008

Subgroups

subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265597);