Finite Group Databasebeta

← Prev (#7635364) Next (#7635366) →

Cayley diagram

C2×#38076 #7635365

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#38076

Subgroups

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

12328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

(1,2);(4,6)(7,8)(9,13)(10,11)(12,14);(8,11)(12,13)(16,19)(17,21);(3,4,5,6)(15,16,18,19)(17,20,21,22);(7,9)(8,12)(10,14)(11,13);(7,8)(9,12)(10,11)(13,14)(15,17)(16,20)(18,21)(19,22)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7849701, #72232096, #72232113, …
maximal normal subgroup ofC2×#7849701, #72232096, #72232113, …
maximal quotient of#72232125, #72232151, C2×#1262353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542561);