Finite Group Databasebeta

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Cayley diagram

C2×#28177 #6462891

order512
nilpotency class3
isoclinismisoclinic class of #24283
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#28177

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2530 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)(5,9)(6,7)(8,10)(11,12)(13,17)(14,18)(15,19)(16,20)(21,26)(22,24)(23,25);(4,7)(8,9)(11,13,15,23,16,22,14,21)(12,17,19,25,20,24,18,26);(12,18)(13,21)(14,15)(17,24)(19,20)(22,23);(3,5,6,10)(4,8,7,9)(13,22)(17,24)(21,23)(25,26)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238928, C2×#6467260, #72239151
maximal normal subgroup of#72238928, C2×#6467260, #72239151
maximal quotient of#72239151

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370087);