Finite Group Databasebeta

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Cayley diagram

C2×#45482 #7702043

order512
nilpotency class2
isoclinismisoclinic class of #45438
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#45482

Subgroups

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

7850 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)(17,18)(19,20)(21,22);(8,11)(9,13)(16,20)(18,21);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,21)(18,20)(19,22);(7,9)(8,12)(10,13)(11,14)(15,18,19,21)(16,17,20,22);(7,8)(9,12)(10,11)(13,14)(15,17)(16,21)(18,20)(19,22)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7764218, C2×#7971932, C2×#7968600
maximal normal subgroup ofC2×#7764218, C2×#7971932, C2×#7968600

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7609239);