Finite Group Databasebeta

← Prev (#6443529) Next (#6443531) →

Cayley diagram

C2×#26947 #6443530

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#26947

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6444791
maximal normal subgroup ofC2×#6444791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350726);