Finite Group Databasebeta

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Cayley diagram

C2×#25434 #6444289

order512
nilpotency class3
isoclinismisoclinic class of #24289
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#25434

Subgroups

subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226904, #72226912, #72226917
maximal normal subgroup of#72226904, #72226912, #72226917

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351485);