Finite Group Databasebeta

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Cayley diagram

C2×#12594 #492918

order512
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphism?
homologyAbelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C42

Aliases

direct productC2×#12594

Subgroups

subgroupG/H
centerC2×C42C42
commutatorC4C22×C4×C8
Frattini, cosocleC22×C8C24
socleC23C2×#97
prime radicalC24C2×M16
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C2×C42 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

862 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)(5,8)(6,7)(9,10)(11,12,15,21,16,18,23,25,17,22,24,26,14,20,13,19);(4,7)(8,10)(12,18,22,20)(13,23)(15,24)(19,26,25,21);(3,5,6,9)(4,8,7,10)(11,13,14,24,17,23,16,15)(12,19,20,26,22,25,18,21)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#493177, #72216468, C2×#128555, …
maximal normal subgroup ofC2×#493177, #72216468, C2×#128555, …
maximal quotient ofC2×#128208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400114);