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Cayley diagram

C2×#25933 #6447662

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25933

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72281773, C2×#6454747, #72283295, …
maximal normal subgroup of#72281773, C2×#6454747, #72283295, …
maximal quotient of#72304900, #72305516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354858);