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Cayley diagram

C2×#21753 #6374134

order512
nilpotency class3
isoclinismisoclinic class of #17392
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#21753

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#186831, #72238876
maximal normal subgroup ofC2×#186831, #72238876
maximal quotient of#72238889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281330);