Finite Group Databasebeta

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Cayley diagram

C2×#21793 #6374245

order512
nilpotency class3
isoclinismisoclinic class of #17392
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C42

Aliases

direct productC2×#21793

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC23×#121C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

6794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235074, #72235089, C2×#6385971, …
maximal normal subgroup of#72235074, #72235089, C2×#6385971, …
maximal quotient of#72251120, #72252043, #72252044, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281441);