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Cayley diagram

C2×#13432 #499146

order512
nilpotency class4
isoclinismisoclinic class of #13432
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13432

Subgroups

subgroupG/H
center, socleC22#2044
commutator, Frattini, cosocleC2×#41C24
prime radicalC2×#3199C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

1173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311946, #72311949, C2×#501108, …
maximal normal subgroup of#72311946, #72311949, C2×#501108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406342);