Finite Group Databasebeta

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Cayley diagram

C2×#41718 #7639770

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#41718

Subgroups

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

6294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

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, 7546966);