Finite Group Databasebeta

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Cayley diagram

C2×#25920 #6447736

order512
nilpotency class3
isoclinismisoclinic class of #24412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25920

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3282 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);(3,4,7,10)(5,9,6,8)(11,12,16,18)(13,17,21,26)(14,15,23,25)(19,20,22,24);(3,5,7,6)(4,8,10,9)(11,13,16,21)(12,17,18,26)(14,22,23,19)(15,24,25,20);(3,5,7,6)(4,9,10,8)(11,14,15,18,16,23,25,12)(13,19,20,26,21,22,24,17);(3,6,7,5)(4,9,10,8)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271418, #72271422, #72271453
maximal normal subgroup of#72271418, #72271422, #72271453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354932);