Finite Group Databasebeta

← Prev (#7668134) Next (#7668136) →

Cayley diagram

C2×#44248 #7668135

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

Aliases

direct productC2×#44248

Subgroups

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

7840 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)(9,13)(11,14)(15,16,19,18)(17,21,22,20);(4,6)(7,8)(9,14)(10,12)(11,13);(7,8,10,12)(9,11,13,14)(15,17,19,22)(16,20,18,21);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14);(7,9)(8,11)(10,13)(12,14)(15,18,19,16)(17,21,22,20)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7668148, C2×#7697591, C2×#7694939
maximal normal subgroup ofC2×#7668148, C2×#7697591, C2×#7694939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7575331);