Finite Group Databasebeta

← Prev (#499186) Next (#499188) →

Cayley diagram

C2×#13452 #499187

order512
nilpotency class4
isoclinismisoclinic class of #13304
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13452

Subgroups

subgroupG/H
center, socle, prime radicalC24C2×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2);(7,8,11,13)(9,14,10,12)(15,16,18,23)(17,25,20,21)(19,22,27,30)(24,26,29,28);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,18,20)(16,21,23,25)(19,26,27,28)(22,24,30,29);(4,6)(7,10,11,9)(8,12,13,14)(16,22)(17,20)(19,28)(21,29)(23,30)(24,25)(26,27)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499235, #72287653, #72287672, …
maximal normal subgroup ofC2×#499235, #72287653, #72287672, …
maximal quotient ofC2×#153330, C2×#153329, #72322632

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406383);