Finite Group Databasebeta

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Cayley diagram

C2×#18633 #6348209

order512
nilpotency class3
isoclinismisoclinic class of #18618
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18633

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

2986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,8,12)(5,13,16,17)(6,11,7,10)(9,14,18,15)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(3,5,8,16)(4,9,12,18)(6,15,7,14)(10,13,11,17)(22,31)(25,30)(27,34)(29,32);(19,21)(20,24)(22,32)(23,28)(25,34)(26,33)(27,30)(29,31);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,17,18,13)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255405);