Finite Group Databasebeta

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Cayley diagram

C4×#3483 #7625408

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC4×#3483

Subgroups

subgroupG/H
centerC2×C42C24
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC2×#3483C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ G
LECSG ▷ C23 ▷ C1

8768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#62759, #72315196
maximal normal subgroup ofC4×#62759, #72315196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532604);