Finite Group Databasebeta

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Cayley diagram

C2×#22166 #6375897

order512
nilpotency class3
isoclinismisoclinic class of #3192
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#22166

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C2×C4×D4
prime radicalC2×#3035C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2174 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);(4,7)(5,9)(11,12,15,19,16,14,22,18)(13,17,20,25,21,23,26,24);(3,4,6,7)(5,8,9,10)(11,13,15,20,16,21,22,26)(12,17,19,25,14,23,18,24);(4,7)(8,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,25)(21,23)(24,26);(3,5,6,9)(4,8,7,10)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6404044, C2×#6403639, C2×#6378175
maximal normal subgroup ofC2×#6404044, C2×#6403639, C2×#6378175

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283093);