Finite Group Databasebeta

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Cayley diagram

C2×#9550 #202477

order512
nilpotency class3
isoclinismisoclinic class of #471
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#9550

Subgroups

subgroupG/H
centerC22×C4#121
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23#388
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#202513, #72215904, #72215908, …
maximal normal subgroup ofC2×#202513, #72215904, #72215908, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109673);