Finite Group Databasebeta

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Cayley diagram

C2×#43903 #7668656

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#43903

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

8914 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)(7,8,12,10)(9,14,11,13)(17,21)(19,22);(7,9,12,11)(8,13,10,14)(15,16)(17,22)(18,20)(19,21);(3,4)(5,6)(15,16,18,20)(17,19,21,22);(7,10,12,8)(9,13,11,14)(15,16)(17,19)(18,20)(21,22);(7,11,12,9)(8,13,10,14)(15,17)(16,19)(18,21)(20,22)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7800957, C2×#7856019, C2×#7686144
maximal normal subgroup ofC2×#7800957, C2×#7856019, C2×#7686144

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7575852);