Finite Group Databasebeta

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Cayley diagram

C2×#44450 #7668777

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#44450

Subgroups

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

8954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 30 points:

(1,2);(4,6)(7,8,11,10)(9,13,14,12)(15,16)(17,21)(18,26)(19,20)(22,27)(23,24)(25,30)(28,29);(8,10)(12,13)(16,20)(17,23)(18,27)(28,30);(3,4)(5,6)(7,9,11,14)(8,12,10,13)(15,16)(17,24)(18,22)(19,20)(21,23)(25,30)(26,27)(28,29);(7,10,11,8)(9,13,14,12)(15,17,19,23)(16,21,20,24)(18,25,27,29)(22,28,26,30);(15,18)(16,22)(17,25)(19,27)(20,26)(21,28)(23,29)(24,30)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7901829, C2×#7812377, C2×#7694912
maximal normal subgroup ofC2×#7901829, C2×#7812377, C2×#7694912
maximal quotient ofC2×#3868389, C2×#3995501, C2×#772847

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7575973);