Finite Group Databasebeta

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Cayley diagram

C2×#25400 #6444350

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#25400

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7156989, C2×#7157035, C2×#6444387
maximal normal subgroup ofC2×#7156989, C2×#7157035, C2×#6444387

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351546);