Finite Group Databasebeta

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Cayley diagram

C2×#27179 #6450824

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#27179

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

2326 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)(13,17)(14,15)(16,18);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,20,23,26)(21,25,22,24);(12,15)(16,17)(20,24)(21,22)(25,26);(4,8)(7,9)(11,13,14,18)(12,16,15,17)(19,21,23,22)(20,25,26,24)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6452465, #72315901, #72315928, …
maximal normal subgroup ofC2×#6452465, #72315901, #72315928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358020);