Finite Group Databasebeta

← Prev (#7705112) Next (#7705114) →

Cayley diagram

C2×#49565 #7705113

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#49565

Subgroups

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

8844 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);(3,4,5,6)(7,8)(9,10)(11,12)(13,14);(8,12)(9,13)(16,19)(17,21);(3,4)(5,6)(15,16,18,19)(17,20,21,22);(7,9,11,13)(8,10,12,14)(15,16)(17,20)(18,19)(21,22);(7,10)(8,13)(9,12)(11,14)(15,17)(16,20)(18,21)(19,22)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7856019, C2×#7733278, C2×#7851307
maximal normal subgroup ofC2×#7856019, C2×#7733278, C2×#7851307

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7612309);