Finite Group Databasebeta

← Prev (#513507) Next (#513509) →

Cayley diagram

C4×#2353 #513508

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C26×C42

Aliases

direct productC4×#2353

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C23×C42
Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

4146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303671, #72303679
maximal normal subgroup of#72303671, #72303679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420704);