Finite Group Databasebeta

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Cayley diagram

C2×#48765 #7703818

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#48765

Subgroups

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

6112 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);(11,12,16,14)(13,15,18,17)(19,20,23,22)(21,25,26,24);(3,4)(5,9)(6,10)(7,8)(11,13)(12,17)(14,15)(16,18)(19,20,23,22)(21,24,26,25);(4,8)(5,10)(12,14)(13,18)(19,21,23,26)(20,24,22,25);(3,5)(4,6)(7,10)(8,9)(11,14,16,12)(13,15,18,17)(19,20,23,22)(21,24,26,25);(3,6)(4,5)(7,9)(8,10)(11,15)(12,13)(14,18)(16,17)(19,22,23,20)(21,25,26,24)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10320757
maximal normal subgroup ofC2×#10320757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7611014);