Finite Group Databasebeta

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Cayley diagram

C22×#1391 #492889

order512
nilpotency class4
isoclinismisoclinic class of AΣL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC22×#1391

Subgroups

subgroupG/H
centerC22×C4#99
commutatorC23C22×C42
Frattini, cosocleC2×#41C24
socleC23#342
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

2446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2);(3,4);(5,6)(7,15)(8,19,20,12)(9,13)(10,14)(11,16,23,29)(17,27)(18,28)(21,32,33,25)(22,26)(24,30,34,36)(31,35);(5,7,9,17,10,18,22,31)(6,11,13,24,14,23,26,34)(8,16,21,30,20,29,33,36)(12,15,25,27,19,28,32,35)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267920, C2×#494380, C2×#146653, …
maximal normal subgroup of#72267920, C2×#494380, C2×#146653, …
maximal quotient of#72267924, #72267929, C2×#146461

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400085);