Finite Group Databasebeta

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Cayley diagram

C4×#1311 #126517

order512
nilpotency class2
isoclinismisoclinic class of D4
automorphism?
homologyAbelianization (H1) = C4×C82 · Schur multiplier (H2) = C43

Aliases

direct productC4×#1311

Subgroups

subgroupG/H
centerC42×C8C22
commutatorC2C4×C82
Frattini, cosocleC2×C4×C8C23
socle, prime radicalC23C2×C4×C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

Frattini subgroup of#72234265, #72234269
maximal subgroup of#72234224, #72234227, #72234237
maximal normal subgroup of#72234224, #72234227, #72234237
maximal quotient of#72234228, #72234239, C4×#7052

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33713);