Finite Group Databasebeta

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Cayley diagram

C2×#37693 #7634950

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#37693

Subgroups

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

11986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7967779, #72233693, #72233710, …
maximal normal subgroup ofC2×#7967779, #72233693, #72233710, …
maximal quotient of#72233693, C2×#648435, C2×#688968, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542146);