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Cayley diagram

C2×#13437 #499123

order512
nilpotency class4
isoclinismisoclinic class of #13436
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13437

Subgroups

subgroupG/H
center, socleC22#2044
commutator, Frattini, cosocleC2×#41C24
prime radicalC2×#535C22
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

1121 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);(3,4,9,14)(5,15,19,32)(6,10,8,12)(7,21,22,13)(11,26,28,18)(16,30,17,31)(20,29,23,25)(24,33,27,34);(4,10)(5,16)(6,8)(7,22)(12,14)(13,29)(17,19)(18,33)(21,25)(24,27)(26,34)(30,31);(3,5,6,17,9,19,8,16)(4,11,12,27,14,28,10,24)(7,18,20,34,22,26,23,33)(13,15,29,30,21,32,25,31)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72312701, #72312719, C2×#501198, …
maximal normal subgroup of#72312701, #72312719, C2×#501198, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406319);