Finite Group Databasebeta

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Cayley diagram

C2×#10016 #233566

order512
nilpotency class3
isoclinismisoclinic class of #10012
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#10016

Subgroups

subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC2×C42C24
socleC23#392
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303252, #72303270, #72303288, …
maximal normal subgroup of#72303252, #72303270, #72303288, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140762);