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Cayley diagram

C2×#42984 #7641650

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#42984

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

9282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7654610, C2×#7852139
maximal normal subgroup ofC2×#7654610, C2×#7852139
maximal quotient ofC2×#988863

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548846);