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Cayley diagram

D4×#362 #497396

order512
nilpotency class4
isoclinismisoclinic class of #481415
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productD4×#362

Subgroups

subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D4SD16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4);(2,4);(5,6,9,13,10,15,25,14)(7,16,18,29,19,33,24,32)(8,20,22,12,23,30,36,31)(11,21,26,17,28,34,27,35);(5,7,10,19)(6,11,15,28)(8,21,23,34)(9,24,25,18)(12,29,31,32)(13,27,14,26)(16,20,33,30)(17,22,35,36)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×#2180, D4×#2148, #72309623, …
maximal normal subgroup ofD4×#2180, D4×#2148, #72309623, …
maximal quotient of#72309623, #72309863

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404592);