Finite Group Databasebeta

← Prev (#7670050) Next (#7670052) →

Cayley diagram

C2×#45417 #7670051

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

Aliases

direct productC2×#45417

Subgroups

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

8854 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)(9,13)(10,11)(12,14)(15,16,19,20)(17,21,22,18);(7,9,11,14)(8,12,10,13)(16,20)(18,21);(3,4,5,6)(9,14)(12,13)(15,17)(16,18)(19,22)(20,21);(7,9)(8,13)(10,12)(11,14)(15,18)(16,17)(19,21)(20,22);(7,10,11,8)(9,12,14,13)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7851305, C2×#7850169
maximal normal subgroup ofC2×#7851305, C2×#7850169

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577247);