Finite Group Databasebeta

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Cayley diagram

C2×#25992 #6452291

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27

Aliases

direct productC2×#25992

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2218 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)(11,12,14,16)(13,17,18,15);(3,5,6,10)(4,7,8,9)(19,20,22,25)(21,23,26,24);(11,13,14,18)(12,15,16,17)(20,23)(21,26)(24,25);(4,8)(7,9)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6452488, C2×#7567084, C2×#7555093
maximal normal subgroup ofC2×#6452488, C2×#7567084, C2×#7555093

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359487);