Finite Group Databasebeta

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Cayley diagram

C2×#31606 #7505224

order512
nilpotency class3
isoclinismisoclinic class of #30396
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27

Aliases

direct productC2×#31606

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

2102 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)(13,15)(14,16)(17,18);(3,5,8,4)(6,10,7,9)(11,13,14,17)(12,15,16,18)(19,20,22,25)(21,23,26,24);(3,6,8,7)(4,9,5,10)(12,16)(15,18)(20,23)(21,26)(24,25);(3,7,8,6)(4,9,5,10)(12,16)(15,18)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6715686, C2×#6695622
maximal normal subgroup ofC2×#6715686, C2×#6695622

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7412420);