Finite Group Databasebeta

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Cayley diagram

C2×#18554 #6348104

order512
nilpotency class3
isoclinismisoclinic class of #18524
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18554

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

4304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72279203, C2×#6362966, #72280108, …
maximal normal subgroup of#72279203, C2×#6362966, #72280108, …
maximal quotient ofC2×#209860, C2×#241691, C22×#18554

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255300);