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Cayley diagram

Q8×#397 #4307467

order512
nilpotency class2
isoclinismisoclinic class of D4×#392
automorphismAut(G) order 12582912
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productQ8×#397

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

3294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 256

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);(13,14,15,16)(18,21)(22,23);(10,12)(17,18)(19,23)(20,21)(22,24);(9,10)(11,12)(13,14)(15,16)(17,19,20,24)(18,22,21,23)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofQ8×#2736
maximal normal subgroup ofQ8×#2736
maximal quotient of#72267675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4214663);