Finite Group Databasebeta

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Cayley diagram

C4×#2213 #495830

order512
nilpotency class4
isoclinismisoclinic class of #2212
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×#2213

Subgroups

subgroupG/H
centerC2×C4#459
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×#459
prime radicalC2×#2213C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

774 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);(6,9)(7,12)(8,10)(13,14)(15,20)(16,19)(17,18);(5,6,10,13,11,14,8,9)(7,12,18,20,16,19,15,17);(5,7)(6,12)(8,15)(9,17)(10,18)(11,16)(13,20)(14,19)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72216353, #72216402, #72216405, …
maximal normal subgroup of#72216353, #72216402, #72216405, …
maximal quotient of#72216276, AGL(1,𝔽5)×#2213, C2×C4×#2213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403026);