Finite Group Databasebeta

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Cayley diagram

C2×#25919 #6447735

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

Aliases

direct productC2×#25919

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3066 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)(13,17)(14,23)(15,21)(16,20)(18,24)(19,26)(22,25);(3,5,7,6)(4,8,10,9)(11,13,16,18)(12,17,20,24)(14,19,25,21)(15,23,26,22);(3,5,7,6)(4,9,10,8)(11,14,15,24,16,25,26,17)(12,18,19,23,20,13,21,22);(3,6,7,5)(4,9,10,8)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271446, #72271452, #72271575
maximal normal subgroup of#72271446, #72271452, #72271575

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354931);