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Cayley diagram

C2×#13374 #498975

order512
nilpotency class4
isoclinismisoclinic class of #13374
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13374

Subgroups

subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

944 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)(5,13,15,30)(6,18)(7,22)(8,11)(9,16,14,32)(10,20)(12,24)(17,26,23,31)(19,28,34,25)(21,29)(27,33);(3,4)(5,14,15,9)(6,10)(7,12)(8,11)(13,16,30,32)(17,28,23,25)(18,20)(19,31,34,26)(21,27)(22,24)(29,33);(3,5)(4,9)(6,19)(7,23)(8,15)(10,26)(11,14)(12,28)(13,29)(16,33)(17,24)(18,30)(20,32)(21,34)(22,25)(27,31)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499269, #72231524, #72231541, …
maximal normal subgroup ofC2×#499269, #72231524, #72231541, …
maximal quotient of#72250194, #72251022, #72251049, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406171);