Finite Group Databasebeta

← Prev (#354792) Next (#354794) →

Cayley diagram

C4×#1781 #354793

order512
nilpotency class3
isoclinismisoclinic class of #1462
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4

Aliases

direct productC4×#1781

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
socleC23C22×C22⋊C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

6118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72236632, AGL(1,𝔽5)×#1781, C4×#12687, …
maximal normal subgroup of#72236632, C4×#12687, C4×#32898, …
maximal quotient ofAGL(1,𝔽5)×#1781, C4×#12687, C4×#12705, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261989);