Finite Group Databasebeta

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Cayley diagram

C2×#9580 #202979

order512
nilpotency class3
isoclinismisoclinic class of #1665
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×#9580

Subgroups

subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#388
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,15,19,16,20,14,17)(13,21,24,25,23,18,22,26);(4,7)(8,9)(12,17,20,19)(18,25,21,26);(4,8)(6,10)(7,9)(11,13,14,22,16,23,15,24)(12,18,19,26,20,21,17,25)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#203002, #72216045, #72216055
maximal normal subgroup ofC2×#203002, #72216045, #72216055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110175);