Finite Group Databasebeta

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Cayley diagram

C2×#25745 #6443301

order512
nilpotency class3
isoclinismisoclinic class of #24464
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25745

Subgroups

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

5194 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)(7,8,12,16)(9,13,18,21)(10,19,17,22)(11,15,20,14);(3,4,6,5)(7,9)(8,13)(10,20)(11,17)(12,18)(14,19)(15,22)(16,21);(7,10,11,9,12,17,20,18)(8,13,14,19,16,21,15,22);(3,5,6,4)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72312814, C2×#6443305
maximal normal subgroup of#72312814, C2×#6443305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350497);