Finite Group Databasebeta

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Cayley diagram

C2×#21767 #6374154

order512
nilpotency class3
isoclinismisoclinic class of #3192
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#21767

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×#121C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

6602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4,5,6)(7,8)(9,13)(10,12)(11,14)(15,16)(17,20)(18,21)(19,22);(7,9)(8,11)(10,14)(12,13)(15,17,19,18)(16,20,22,21);(9,14)(11,13)(16,21)(17,18)(20,22);(8,12)(11,13)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72292860, C2×#6374218
maximal normal subgroup of#72292860, C2×#6374218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281350);