Finite Group Databasebeta

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Cayley diagram

C2×#18192 #6346460

order512
nilpotency class3
isoclinismisoclinic class of #18192
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18192

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(7,10)(11,12)(13,19)(14,22)(15,17)(16,21)(18,24)(20,26)(23,25);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,23)(15,20)(17,25)(18,26)(19,22)(21,24);(3,5)(4,7)(6,9)(8,10);(12,17)(13,20)(18,22)(21,23)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346594, #72284431, #72284450, …
maximal normal subgroup ofC2×#6346594, #72284431, #72284450, …
maximal quotient of#72285719, #72285778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253656);