Finite Group Databasebeta

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Cayley diagram

C4×#3361 #6384500

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC4×#3361

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#459C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,3,4);(6,9)(10,11)(13,14,17,21)(15,18,23,27)(16,19,24,20)(22,25,28,26);(5,6,8,9)(7,11,12,10)(13,15,17,23)(14,18,21,27)(16,22,24,28)(19,25,20,26);(14,19)(16,24)(18,25)(20,21)(22,28)(26,27);(5,7,8,12)(6,10,9,11)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282287
maximal normal subgroup of#72282287

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291696);