Finite Group Databasebeta

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Cayley diagram

C2×#39212 #7637042

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#39212

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#8698849, C2×#8341659, C2×#7969247
maximal normal subgroup ofC2×#8698849, C2×#8341659, C2×#7969247

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7544238);