Finite Group Databasebeta

← Prev (#7665628) Next (#7665630) →

Cayley diagram

Q8×#540 #7665629

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

Aliases

direct productQ8×#540

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#540C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

5244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572825);