Finite Group Databasebeta

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Cayley diagram

C2×#14881 #607861

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#14881

Subgroups

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

5958 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);(4,6)(7,8,9,10)(11,12,15,13)(14,17,18,16)(20,22)(21,25);(12,13)(16,17)(19,20,23,22)(21,24,25,26);(7,8)(9,10)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(7,8)(9,10)(11,14,15,18)(12,16,13,17)(19,22,23,20)(21,26,25,24)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#1421181, #72267034, #72267037, …
maximal normal subgroup ofC2×#1421181, #72267034, #72267037, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515057);