Finite Group Databasebeta

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Cayley diagram

C2×#19501 #6354573

order512
nilpotency class3
isoclinismisoclinic class of #19456
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#19501

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3106C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261769);