Finite Group Databasebeta

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Cayley diagram

C2×#41300 #7638987

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

Aliases

direct productC2×#41300

Subgroups

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

6614 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)(5,6)(7,8)(9,10)(19,20,24,22)(21,26,23,25);(4,8)(6,9)(11,12)(13,17)(14,15)(16,18)(21,23)(25,26);(4,8)(5,10)(12,15)(16,17)(19,21,24,23)(20,25,22,26);(3,5)(4,9)(6,8)(7,10)(11,12)(13,16)(14,15)(17,18)(19,22,24,20)(21,25,23,26);(3,6,7,9)(4,5,8,10)(11,13)(12,16)(14,18)(15,17)(19,23,24,21)(20,25,22,26)

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, 7546183);