Finite Group Databasebeta

← Prev (#8092021) Next (#8092023) →

Cayley diagram

Q8×#564 #8092022

order512
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productQ8×#564

Subgroups

subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

4089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(11,17)(12,20)(14,18)(15,22)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(9,10,13,16)(11,18,17,14)(12,15,20,22)(19,24,23,21)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999218);