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Cayley diagram

C2×#28819 #6465750

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28819

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6583220
maximal normal subgroup ofC2×#6583220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372946);