Finite Group Databasebeta

← Prev (#6432578) Next (#6432580) →

Cayley diagram

C2×#23794 #6432579

order512
nilpotency class3
isoclinismisoclinic class of #2010
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#23794

Subgroups

subgroupG/H
centerC2×C8#127
commutator, Frattini, cosocleC22×C4C25
socleC22#2397
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,7,11,8,12,20,26)(5,13,15,21,16,25,19,9)(6,10,17,23,18,24,31,34)(14,27,28,32,29,33,30,22);(3,5)(4,9)(6,14)(7,19)(8,16)(10,22)(11,25)(12,21)(13,26)(15,20)(17,30)(18,29)(23,33)(24,32)(27,34)(28,31);(9,21)(13,25)(22,32)(27,33);(3,6,7,17,8,18,20,31)(4,10,11,23,12,24,26,34)(5,14,19,30,16,29,15,28)(9,22,25,33,21,32,13,27)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6469085, #72250365, #72250375, …
maximal normal subgroup ofC2×#6469085, #72250365, #72250375, …
maximal quotient of#72256923, #72257191, #72283833, …
automorphism group ofC4×C32, #1397
outer automorphism group ofC4×C32, #7331, #7465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339775);