Finite Group Databasebeta

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Cayley diagram

C2×#20558 #6370282

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C29

Aliases

direct productC2×#20558

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C23×C42
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3486 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);(7,8)(9,13)(10,12)(11,14)(15,16,19,21,18,17,20,22);(3,4,5,6)(7,9)(8,11)(10,14)(12,13)(16,17)(21,22);(9,14)(11,13)(15,17,19,22,18,16,20,21);(8,12)(11,13)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6370380, #72257830, #72257834, …
maximal normal subgroup ofC2×#6370380, #72257830, #72257834, …
maximal quotient of#72239465, C2×#493318

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277478);