Finite Group Databasebeta

← Prev (#7837) Next (#7839) →

Cayley diagram

C4×C82 #7838

order256
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) order 786432
homologyAbelianization (H1) = G · Schur multiplier (H2) = C42×C8

Aliases

abelianC4×C82

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C42C23
socle, prime radicalC23C2×C42
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

center of#72234228, C4×#7052, #72234260, …
Frattini subgroup of#72234231, #72234251, #72234261, …
Fitting subgroup ofC4×C82⋊C3
maximal subgroup of#93257, #93264, C83, …
maximal normal subgroup of#93257, #93264, C83, …
maximal quotient ofC83, C8×#322, #94852, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 826);