Finite Group Databasebeta

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Cayley diagram

C2×#27310 #6440876

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

Aliases

direct productC2×#27310

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2184 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)(14,18)(15,17);(4,7)(8,10)(11,13,15,14)(12,16,17,18)(20,23)(21,25)(24,26);(3,5,6,9)(4,8,7,10)(12,17)(16,18)(19,20)(21,26)(22,24)(23,25);(4,7)(8,10)(11,14,15,13)(12,18,17,16)(20,24)(23,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6695188, C2×#6442082, C2×#6699654
maximal normal subgroup ofC2×#6695188, C2×#6442082, C2×#6699654

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348072);