Finite Group Databasebeta

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Cayley diagram

C2×#59419 #7666391

order512
nilpotency class2
isoclinismisoclinic class of #58990
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#59419

Subgroups

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

10622 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);(4,5)(7,8,11,9)(10,13,14,12)(15,16,18,19)(17,21,22,20);(4,5)(8,9)(12,13)(16,19)(20,21);(3,4,6,5)(15,16)(17,20)(18,19)(21,22);(3,5,6,4)(7,9,11,8)(10,13,14,12)(15,16)(17,20)(18,19)(21,22);(7,10,11,14)(8,12,9,13)(15,17,18,22)(16,20,19,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261462, C2×#7699554, #72261555, …
maximal normal subgroup of#72261462, C2×#7699554, #72261555, …
maximal quotient of#72303415, #72303417

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573587);