Finite Group Databasebeta

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Cayley diagram

Q8×#444 #6385144

order512
nilpotency class3
isoclinismisoclinic class of D4×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productQ8×#444

Subgroups

subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalC2×#135C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72309671, SD16×#444, #72309692, …
maximal normal subgroup of#72309671, SD16×#444, #72309692, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292340);