Finite Group Databasebeta

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Cayley diagram

C2×#14105 #514433

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42

Aliases

direct productC2×#14105

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutatorC23C24×C4
Frattini, cosocleC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

3794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#534011, C2×#1101074, C2×#2816035
maximal normal subgroup ofC2×#534011, C2×#1101074, C2×#2816035
maximal quotient ofC2×#154771

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421629);