Finite Group Databasebeta

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Cayley diagram

C2×#22717 #6380251

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#22717

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C4⋊C4
prime radicalC23×D4Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2444 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);(4,7)(8,10)(11,12,15,18,16,19,14,24)(13,20,22,25,23,26,21,17);(3,4)(5,8)(6,7)(9,10);(3,5,6,9)(4,8,7,10)(11,13,14,21,16,23,15,22)(12,17,18,20,19,25,24,26);(4,7)(8,10)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6386772, C2×#6384166, C2×#6403639
maximal normal subgroup ofC2×#6386772, C2×#6384166, C2×#6403639

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287447);