Finite Group Databasebeta

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Cayley diagram

C2×#42732 #7641204

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#42732

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7659759, C2×#8267284, C2×#8323171
maximal normal subgroup ofC2×#7659759, C2×#8267284, C2×#8323171
maximal quotient ofC2×#689120

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548400);