Finite Group Databasebeta

← Prev (#7664860) Next (#7664862) →

Cayley diagram

D4×#540 #7664861

order512
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productD4×#540

Subgroups

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

8542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72263769, #72263771
maximal normal subgroup of#72263769, #72263771
maximal quotient ofD4×#2328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572057);