Finite Group Databasebeta

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Cayley diagram

C2×#55579 #8100608

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#55579

Subgroups

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

6674 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);(3,4,7,8)(5,6,9,10)(11,12,14,17)(13,16,18,15)(19,20)(21,23)(22,25)(24,26);(4,8)(5,9)(11,13,14,18)(12,15,17,16)(20,21)(24,25);(13,18)(15,16)(19,20,23,21)(22,24,26,25);(3,5)(4,6)(7,9)(8,10)(11,13)(12,16)(14,18)(15,17)(19,21,23,20)(22,25,26,24);(3,6)(4,5)(7,10)(8,9)(11,12)(13,16)(14,17)(15,18)(19,22,23,26)(20,24,21,25)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10442541
maximal normal subgroup ofC2×#10442541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8007804);