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Cayley diagram

C2×#25826 #6445792

order512
nilpotency class3
isoclinismisoclinic class of #24412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#25826

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271523, #72271535, #72271570
maximal normal subgroup of#72271523, #72271535, #72271570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352988);