Finite Group Databasebeta

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Cayley diagram

D8×2−1+4 #10584675

order512
nilpotency class3
isoclinismisoclinic class of D8×H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214

Aliases

direct productD8×2−1+4

Subgroups

subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×2−1+4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

8114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD8×#585
maximal normal subgroup ofD8×#585
maximal quotient of#72320055, #72320269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10491871);