Finite Group Databasebeta

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Cayley diagram

C2×#62245 #10577526

order512
nilpotency class3
isoclinismisoclinic class of #62243
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214

Aliases

direct productC2×#62245

Subgroups

subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

9098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72313121, #72313129, C2×D8×H(5,𝔽2), …
maximal normal subgroup of#72313121, #72313129, C2×D8×H(5,𝔽2), …
maximal quotient of#72314254, #72314396, #72315178

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10484722);