Finite Group Databasebeta

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Cayley diagram

C2×#32050 #7507532

order512
nilpotency class3
isoclinismisoclinic class of #30396
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#32050

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2324 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);(3,4,7,10)(5,6,9,8)(11,12,14,16)(13,17,18,15);(3,5,7,9)(4,8,10,6)(11,13,14,18)(12,15,16,17)(19,20,22,25)(21,23,26,24);(3,6,7,8)(4,9,10,5)(12,16)(15,17)(20,23)(21,26)(24,25);(4,10)(5,9)(13,18)(15,17)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7512280, C2×#6695558, C2×#6724588
maximal normal subgroup ofC2×#7512280, C2×#6695558, C2×#6724588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414728);