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Cayley diagram

C2×#49381 #7704356

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#49381

Subgroups

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

6364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7611552);