Finite Group Databasebeta

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Cayley diagram

C2×#26408 #6441895

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#26408

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

2662 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)(11,12,14,15)(13,17,18,16);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,20)(21,23)(22,25)(24,26);(11,13,14,18)(12,16,15,17)(20,23)(21,26)(24,25);(4,8)(7,9)(12,15)(16,17)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6452488, C2×#7572018, C2×#7548479
maximal normal subgroup ofC2×#6452488, C2×#7572018, C2×#7548479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349091);