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Cayley diagram

C2×C4⋊#401 #608399

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×C4⋊#401

Subgroups

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

3230 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);(4,6)(8,10)(12,13)(15,16,20,18)(17,22,19,21)(23,24,27,30)(25,28,29,26);(7,8,11,10)(9,13,14,12)(15,17,20,19)(16,21,18,22)(23,25,27,29)(24,28,30,26);(7,9,11,14)(8,12,10,13)(15,18,20,16)(17,22,19,21)(23,24)(25,28)(26,29)(27,30);(3,4,5,6)(7,10,11,8)(9,13,14,12)(15,19,20,17)(16,22,18,21)(23,26,27,28)(24,29,30,25)

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, 515595);