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Cayley diagram

Q8×#390 #519053

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4

Aliases

direct productQ8×#390

Subgroups

subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

5738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofQ8×#2679
maximal normal subgroup ofQ8×#2679
maximal quotient of#72267414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426249);