Finite Group Databasebeta

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Cayley diagram

C2×#15055 #608140

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#15055

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

3446 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)(8,11)(9,13)(15,16,19,17)(18,20,22,21)(24,27)(25,29);(3,4,6,5)(16,17)(20,21)(23,24)(25,28)(26,27)(29,30);(3,5,6,4)(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,21,22,20)(23,25,26,29)(24,28,27,30);(7,9,10,13)(8,12,11,14)(15,18,19,22)(16,20,17,21)(23,24)(25,28)(26,27)(29,30)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515336);