Finite Group Databasebeta

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Cayley diagram

C2×#30327 #6433859

order512
nilpotency class3
isoclinismisoclinic class of #29742
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#30327

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6699142, C2×#6729941, C2×#6434019
maximal normal subgroup ofC2×#6699142, C2×#6729941, C2×#6434019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341055);