Finite Group Databasebeta

← Prev (#6629414) Next (#6629416) →

Cayley diagram

C2×#33033 #6629415

order512
nilpotency class3
isoclinismisoclinic class of #32878
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#33033

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536611);