Finite Group Databasebeta

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Cayley diagram

C2×#35289 #7628558

order512
nilpotency class2
isoclinismisoclinic class of #34980
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C211×C42

Aliases

direct productC2×#35289

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC22×#532C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

9826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72302319, C2×#7671592, C2×#7641983
maximal normal subgroup of#72302319, C2×#7671592, C2×#7641983

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535754);