Finite Group Databasebeta

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Cayley diagram

C2×#23596 #6432098

order512
nilpotency class3
isoclinismisoclinic class of #2021
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#23596

Subgroups

subgroupG/H
centerC22×C4#120
commutator, Frattini, cosocleC24C25
socleC23C2×#120
prime radicalC2×#3459C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

3976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6432099, C2×#6454043, C2×#6454042
maximal normal subgroup ofC2×#6432099, C2×#6454043, C2×#6454042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339294);