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Cayley diagram

C2×#24145 #6448655

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24145

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3348 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)(5,9)(6,7)(8,10)(11,12,15,18)(13,16,20,25)(14,22,23,17)(19,26,21,24);(4,7)(8,9)(11,13)(12,16)(14,19)(15,20)(17,24)(18,25)(21,23)(22,26);(13,19)(14,23)(16,24)(17,22)(20,21)(25,26);(3,5,6,10)(4,8,7,9)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6448661, C2×#6453883, C2×#6453870
maximal normal subgroup ofC2×#6448661, C2×#6453883, C2×#6453870
maximal quotient ofC2×#258132, #72309297

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355851);