Finite Group Databasebeta

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Cayley diagram

C2×#36285 #7629935

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#36285

Subgroups

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

5346 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);(11,12,15,18)(13,16,14,17)(20,24)(25,26);(3,4,6,8)(5,7,10,9)(11,12)(13,16)(14,17)(15,18)(19,20)(21,25)(22,26)(23,24);(3,4)(5,9)(6,8)(7,10)(11,12)(13,16)(14,17)(15,18)(19,21,23,22)(20,25,24,26);(3,5)(4,7)(6,10)(8,9)(11,13,15,14)(12,16,18,17)(19,21,23,22)(20,25,24,26);(3,4)(5,9)(6,8)(7,10)(11,14,15,13)(12,17,18,16)(19,22,23,21)(20,26,24,25)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537131);