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Cayley diagram

Q8×#546 #7699516

order512
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productQ8×#546

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#546C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

8353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72304587, #72304614, #41×#546
maximal normal subgroup of#72304587, #72304614, #41×#546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606712);