Finite Group Databasebeta

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Cayley diagram

C2×#27333 #6449430

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#27333

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2174 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)(5,10)(6,8)(7,9)(11,12)(13,18)(14,17)(15,16)(20,23)(24,26);(3,5,7,6)(4,8,9,10)(12,16)(17,18)(20,24)(22,25)(23,26);(3,6,7,5)(4,8,9,10)(11,13,15,14)(12,17,16,18)(19,20,21,23)(22,26,25,24);(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×#6455230, C2×#6742022
maximal normal subgroup ofC2×#6455230, C2×#6742022

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356626);