Finite Group Databasebeta

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Cayley diagram

C2×#21660 #6373525

order512
nilpotency class3
isoclinismisoclinic class of #3055
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#21660

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2130 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)(5,8)(6,7)(9,10)(11,12,15,19,16,20,24,18)(13,21,14,17,22,25,23,26);(4,7)(8,10)(11,13,15,14,16,22,24,23)(12,17,18,21,20,26,19,25);(3,5,6,9)(4,8,7,10)(11,13,15,14,16,22,24,23)(12,17,18,21,20,26,19,25);(11,14,15,22,16,23,24,13)(12,17,19,25,20,26,18,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6373967, C2×#6384166, C2×#6403639
maximal normal subgroup ofC2×#6373967, C2×#6384166, C2×#6403639

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280721);