Finite Group Databasebeta

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Cayley diagram

C2×#32893 #6628865

order512
nilpotency class3
isoclinismisoclinic class of #32888
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C212

Aliases

direct productC2×#32893

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#585C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

17068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6628866, #72259221, #72267278
maximal normal subgroup ofC2×#6628866, #72259221, #72267278
maximal quotient of#72259221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536061);