Finite Group Databasebeta

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Cayley diagram

C2×#27360 #6450271

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27

Aliases

direct productC2×#27360

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1962 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);(3,4,7,9)(5,10,6,8)(11,12,15,16)(13,18,14,17);(3,5,7,6)(4,8,9,10)(12,16)(17,18)(20,23)(21,25)(24,26);(3,6,7,5)(4,8,9,10)(11,13,15,14)(12,17,16,18)(19,20,22,24)(21,26,25,23);(4,9)(8,10)(11,14,15,13)(12,18,16,17)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6715686, C2×#6455230, C2×#6729734
maximal normal subgroup ofC2×#6715686, C2×#6455230, C2×#6729734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357467);