Finite Group Databasebeta

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Cayley diagram

C2×#46006 #7702675

order512
nilpotency class2
isoclinismisoclinic class of #45438
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#46006

Subgroups

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

4990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

(1,2);(4,6)(8,9)(13,17)(16,18)(19,20,22,24)(21,25,26,23);(3,4,7,6)(5,9,10,8)(12,15)(13,17);(11,12)(13,16)(14,15)(17,18)(19,21,22,26)(20,23,24,25);(3,5,7,10)(4,8,6,9)(11,13,14,17)(12,16,15,18);(3,6,7,4)(5,9,10,8)(11,12)(13,16)(14,15)(17,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7609871);