Finite Group Databasebeta

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Cayley diagram

C2×#37710 #7635121

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

Aliases

direct productC2×#37710

Subgroups

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

8284 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);(7,8,11,10)(9,13,14,12)(16,17)(20,21);(4,6)(8,10)(12,13)(15,16,19,17)(18,21,22,20);(3,4)(5,6)(15,16,19,17)(18,20,22,21);(7,9,11,14)(8,12,10,13)(15,17,19,16)(18,21,22,20);(3,4)(5,6)(7,10,11,8)(9,13,14,12)(15,18,19,22)(16,20,17,21)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7697590, C2×#7635122, C2×#7698179
maximal normal subgroup ofC2×#7697590, C2×#7635122, C2×#7698179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542317);