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Cayley diagram

C2×#18133 #6346373

order512
nilpotency class3
isoclinismisoclinic class of #18132
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18133

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2404C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4466 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)(16,20)(18,24)(21,26)(23,25);(3,4)(5,7)(6,8)(9,10)(11,13,15,21)(12,16,17,25)(14,23,24,20)(18,26,22,19);(3,5)(4,7)(6,9)(8,10);(12,17)(16,25)(18,22)(19,26)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346594, C2×#249331, #72284024, …
maximal normal subgroup ofC2×#6346594, C2×#249331, #72284024, …
maximal quotient of#72284476, #72284683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253569);