Finite Group Databasebeta

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Cayley diagram

C2×#12062 #483578

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C23×C16 · Schur multiplier (H2) = C26

Aliases

direct productC2×#12062

Subgroups

subgroupG/H
centerC22×C8C2×D4
commutatorC4C23×C16
Frattini, cosocleC4×C8C24
socleC23C8×D4
prime radicalC22×D8C8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225524, C2×#122963, D3×#12062, …
maximal normal subgroup of#72225524, C2×#122963, #72258136, …
maximal quotient of#72225518, C2×#122963, D3×#12062, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390774);