Finite Group Databasebeta

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Cayley diagram

C2×#22395 #6376960

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#22395

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2194 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)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,18)(19,20,24,25,23,21,22,26);(3,5,8,4)(6,9,7,10)(19,21,24,26,23,20,22,25);(3,6,8,7)(4,9,5,10)(12,15)(16,18)(20,21)(25,26);(3,7,8,6)(4,9,5,10)(11,13,14,17)(12,16,15,18)(19,22,23,24)(20,25,21,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6377437, #72257830
maximal normal subgroup ofC2×#6377437, #72257830

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6284156);