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Cayley diagram

C2×#37038 #7634285

order512
nilpotency class2
isoclinismisoclinic class of #36702
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#37038

Subgroups

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

6240 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);(3,4,7,8)(5,10,9,6)(12,16)(14,17)(20,23)(24,26);(4,8)(5,9)(12,16)(13,18);(11,12,15,16)(13,17,18,14)(19,20)(21,24)(22,23)(25,26);(3,5,7,9)(4,6,8,10)(11,13)(12,17)(14,16)(15,18);(3,6)(4,9)(5,8)(7,10)(11,14,15,17)(12,13,16,18)(19,21,22,25)(20,24,23,26)

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, 7541481);