Finite Group Databasebeta

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Cayley diagram

C2×#24228 #6442557

order512
nilpotency class3
isoclinismisoclinic class of #23912
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#24228

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4888 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);(4,5)(7,8)(9,15)(10,18)(11,14)(12,17)(13,19)(16,22)(20,21);(3,4,6,5)(10,19)(12,21)(13,18)(17,20);(7,9,11,16)(8,12,14,21)(10,20,19,17)(13,22,18,15);(3,5,6,4)(9,16)(12,21)(15,22)(17,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6442561
maximal normal subgroup ofC2×#6442561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349753);