Finite Group Databasebeta

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Cayley diagram

D4×#443 #6383858

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productD4×#443

Subgroups

subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalD4×#135C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×#2955, #72286208, #72286221, …
maximal normal subgroup ofD4×#2955, #72286208, #72286221, …
maximal quotient of#72286221, #72287239, #72287409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291054);