Finite Group Databasebeta

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Cayley diagram

C2×#45982 #7702633

order512
nilpotency class2
isoclinismisoclinic class of #45438
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#45982

Subgroups

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

6540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#8091786, C2×#8346424, C2×#8605582
maximal normal subgroup ofC2×#8091786, C2×#8346424, C2×#8605582
maximal quotient ofC2×#873746

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7609829);