Finite Group Databasebeta

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Cayley diagram

C2×#23716 #6432352

order512
nilpotency class3
isoclinismisoclinic class of #2018
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#23716

Subgroups

subgroupG/H
centerC22×C4#121
commutator, Frattini, cosocleC22×C4C25
socleC23C2×#121
prime radicalC2×#2018C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2418 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)(11,12,14,18,15,19,23,17)(13,16,20,24,22,26,21,25);(4,7)(8,10)(11,13)(12,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,23);(12,17)(13,20)(14,23)(18,19)(21,22)(24,25);(3,5,6,9)(4,8,7,10)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6467239, C2×#211085, C2×#211082
maximal normal subgroup ofC2×#6467239, C2×#211085, C2×#211082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339548);