Finite Group Databasebeta

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Cayley diagram

C2×#16877 #611656

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#16877

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

3486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

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

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, 518852);