Finite Group Databasebeta

← Prev (#7500743) Next (#7500745) →

Cayley diagram

C2×#30434 #7500744

order512
nilpotency class3
isoclinismisoclinic class of #30434
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#30434

Subgroups

subgroupG/H
center, socleC22#2668
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72215520, #72215721, #72215936, …
maximal normal subgroup of#72215520, #72215721, #72215936, …
maximal quotient of#72215520, C2×#383145, #72238734, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407940);