Finite Group Databasebeta

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Cayley diagram

C2×#32972 #6629105

order512
nilpotency class3
isoclinismisoclinic class of #32888
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#32972

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4122 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);(3,4,6,8)(5,9,10,7)(12,16)(13,17)(18,23)(19,25)(20,22)(24,26);(3,5,6,10)(4,7,8,9)(11,12,15,16)(13,17,20,22)(14,18,21,23)(19,24,26,25);(11,13)(12,17)(14,19)(15,20)(16,22)(18,24)(21,26)(23,25);(11,14,15,21)(12,18,16,23)(13,19,20,26)(17,24,22,25)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268165, #72268166, #72268185
maximal normal subgroup of#72268165, #72268166, #72268185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536301);