Finite Group Databasebeta

← Prev (#7629400) Next (#7629402) →

Cayley diagram

C2×#35689 #7629401

order512
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#35689

Subgroups

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

9450 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)(9,10)(11,13)(12,14)(16,19)(20,22);(3,4)(5,6)(7,9)(8,12)(10,13)(11,14);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22);(7,8)(9,12)(10,14)(11,13)(15,17,18,21)(16,20,19,22);(7,10,11,12)(8,9,13,14)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264734, #72264785, #72264791, …
maximal normal subgroup of#72264734, #72264785, #72264791, …
maximal quotient ofC2×#4306434, #72280169, #72281175, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536597);