Finite Group Databasebeta

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Cayley diagram

C2×#41233 #7638701

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

Aliases

direct productC2×#41233

Subgroups

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

6380 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);(11,12,15,13)(14,16,18,17)(19,20,23,24)(21,22,25,26);(3,4)(5,9)(6,7)(8,10)(11,12,15,13)(14,17,18,16)(20,24)(21,25);(4,7)(8,9)(12,13)(16,17)(19,21)(20,22)(23,25)(24,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,22)(20,21)(23,26)(24,25);(3,5)(4,8)(6,10)(7,9)(11,14,15,18)(12,16,13,17)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#9779192, C2×#10408065, C2×#7649973
maximal normal subgroup ofC2×#9779192, C2×#10408065, C2×#7649973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545897);