Finite Group Databasebeta

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Cayley diagram

C2×#19283 #6353925

order512
nilpotency class3
isoclinismisoclinic class of #19279
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#19283

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

2068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72305433, #72305438, C2×#6354079
maximal normal subgroup of#72305433, #72305438, C2×#6354079

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261121);