Finite Group Databasebeta

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Cayley diagram

C2×#43206 #7667307

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#43206

Subgroups

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

6406 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);(5,9)(8,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,26);(3,4)(5,10)(6,7)(8,9)(11,12,14,16)(13,15,17,18)(19,20)(21,22)(23,24)(25,26);(4,7)(8,10)(20,24)(21,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,25)(20,22,24,26);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,15,16,18)(19,22)(20,25)(21,24)(23,26)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7667313
maximal normal subgroup ofC2×#7667313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574503);