Finite Group Databasebeta

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Cayley diagram

C2×#35827 #7629333

order512
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#35827

Subgroups

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

6450 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);(11,12,16,14)(13,18,15,17)(20,21)(24,25);(3,4)(5,8)(6,9)(7,10)(11,13,16,15)(12,17,14,18)(19,20,23,21)(22,24,26,25);(3,5)(4,7)(6,10)(8,9)(11,12,16,14)(13,18,15,17)(19,21,23,20)(22,25,26,24);(3,5)(4,8)(6,10)(7,9)(11,14,16,12)(13,18,15,17);(3,4)(5,7)(6,9)(8,10)(11,15,16,13)(12,17,14,18)(19,22,23,26)(20,24,21,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7837055, C2×#7837057, C2×#7629435
maximal normal subgroup ofC2×#7837055, C2×#7837057, C2×#7629435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536529);