Finite Group Databasebeta

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Cayley diagram

C2×#36565 #7630270

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

Aliases

direct productC2×#36565

Subgroups

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

5618 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);(3,4,7,5)(6,8,10,9)(19,20,22,24)(21,23,25,26);(3,4,7,5)(6,9,10,8)(11,12,16,14)(13,18,15,17)(19,20)(21,23)(22,24)(25,26);(4,5)(8,9)(11,13,16,15)(12,17,14,18)(19,20)(21,23)(22,24)(25,26);(3,5,7,4)(6,8,10,9)(11,14,16,12)(13,17,15,18)(19,21,22,25)(20,23,24,26);(3,6,7,10)(4,8,5,9)(11,15,16,13)(12,18,14,17)

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, 7537466);