Finite Group Databasebeta

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Cayley diagram

C2×#26768 #6441240

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

Aliases

direct productC2×#26768

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2292 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);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18);(4,7)(8,10)(20,23)(21,25)(24,26);(3,5,6,9)(4,8,7,10)(12,15)(16,18)(19,20)(21,26)(22,24)(23,25);(4,7)(8,10)(11,13,14,17)(12,16,15,18)(20,24)(23,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6741717, C2×#6442082, C2×#6729798
maximal normal subgroup ofC2×#6741717, C2×#6442082, C2×#6729798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348436);