Finite Group Databasebeta

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Cayley diagram

C2×#47700 #7701650

order512
nilpotency class2
isoclinismisoclinic class of #46350
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#47700

Subgroups

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

6090 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);(11,12)(13,18)(14,17)(15,16)(21,25)(23,26);(3,4)(5,6)(7,8)(9,10)(12,16)(17,18)(19,20,22,24)(21,23,25,26);(4,8)(5,9)(11,13,15,14)(12,17,16,18)(19,21)(20,23)(22,25)(24,26);(3,5,7,9)(4,6,8,10)(11,12)(13,17)(14,18)(15,16);(3,6)(4,9)(5,8)(7,10)(11,14,15,13)(12,18,16,17)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72278796, #72278852, C2×#8934760
maximal normal subgroup of#72278796, #72278852, C2×#8934760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7608846);