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Cayley diagram

C2×#56549 #8099052

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#56549

Subgroups

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

5764 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);(4,7)(8,10)(11,12,16,17)(13,15,14,18);(5,9)(8,10)(12,17)(13,14)(19,20)(21,25)(22,24)(23,26);(3,4,6,7)(5,8,9,10)(11,13,16,14)(12,18,17,15)(19,20,22,24)(21,23,26,25);(3,4)(5,8)(6,7)(9,10)(11,14,16,13)(12,15,17,18)(19,20)(21,23)(22,24)(25,26);(3,5,6,9)(4,8,7,10)(11,15)(12,13)(14,17)(16,18)(19,21)(20,23)(22,26)(24,25)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#9266284
maximal normal subgroup ofC2×#9266284

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8006248);