Finite Group Databasebeta

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Cayley diagram

C2×#8112 #181373

order512
nilpotency class3
isoclinismisoclinic class of #1304
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C27

Aliases

direct productC2×#8112

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#95
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72280897, #72280942, #72280963, …
maximal normal subgroup of#72280897, #72280942, #72280963, …
maximal quotient of#72309224, C2×#128179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88569);