Finite Group Databasebeta

← Prev (#354286) Next (#354288) →

Cayley diagram

C4×#1891 #354287

order512
nilpotency class3
isoclinismisoclinic class of B(4,𝔽2)
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC4×#1891

Subgroups

subgroupG/H
centerC42#120
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C42 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72216090, AGL(1,𝔽5)×#1891, C4×#23882, …
maximal normal subgroup of#72216090, C4×#23882, #72227268, …
maximal quotient ofAGL(1,𝔽5)×#1891, #72257938, #72264034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261483);