Finite Group Databasebeta

← Prev (#6446929) Next (#6446931) →

Cayley diagram

C2×#26684 #6446930

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#26684

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7589657, C2×#7531855, C2×#6452488
maximal normal subgroup ofC2×#7589657, C2×#7531855, C2×#6452488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354126);