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Cayley diagram

Q8×#543 #7665727

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

Aliases

direct productQ8×#543

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

5836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72267892, #72267908
Fitting subgroup of#72267892
maximal subgroup of#72267892, Q8×#2029
maximal normal subgroup of#72267892, Q8×#2029
maximal quotient of#72267604, #72267610, Q8×#2840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572923);