Finite Group Databasebeta

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Cayley diagram

C2×#44445 #7668689

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#44445

Subgroups

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

6728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7668733, C2×#7837049, C2×#7837046
maximal normal subgroup ofC2×#7668733, C2×#7837049, C2×#7837046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7575885);