Finite Group Databasebeta

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Cayley diagram

C2×#43412 #7667643

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

Aliases

direct productC2×#43412

Subgroups

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

6454 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)(11,12,15,13)(14,17,18,16)(19,20)(21,22)(23,25)(24,26);(3,4)(5,10)(6,7)(8,9)(11,12)(13,15)(14,17)(16,18)(19,21,24,23)(20,25,26,22);(4,7)(8,10)(12,13)(16,17);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,22,24,25)(20,21,26,23);(3,5)(4,8)(6,9)(7,10)(11,14)(12,16)(13,17)(15,18)(19,23,24,21)(20,25,26,22)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7850129, C2×#7667657, C2×#7851433
maximal normal subgroup ofC2×#7850129, C2×#7667657, C2×#7851433

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574839);