Finite Group Databasebeta

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Cayley diagram

C2×#45643 #7702023

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

Aliases

direct productC2×#45643

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

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, 7609219);