Finite Group Databasebeta

← Prev (#6348334) Next (#6348336) →

Cayley diagram

C2×#18726 #6348335

order512
nilpotency class3
isoclinismisoclinic class of #18712
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18726

Subgroups

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

2892 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)(5,9)(6,8)(7,10)(11,14)(12,17)(13,15)(16,18)(21,27)(22,29)(24,32)(25,30)(28,31)(33,34);(4,8)(6,14)(9,15)(10,11)(13,18)(16,17)(19,20,23,26)(21,24,28,33)(22,30,29,25)(27,34,31,32);(3,5)(4,9)(6,13)(7,12)(8,15)(10,17)(11,16)(14,18)(19,21)(20,24)(22,31)(23,28)(25,34)(26,33)(27,29)(30,32);(4,10)(5,12)(8,11)(13,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255531);