Finite Group Databasebeta

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Cayley diagram

C2×#28686 #6466234

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28686

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2012 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)(7,8,11,14)(9,13,10,12)(15,16)(17,20)(18,21)(19,22)(23,30)(24,28)(25,27)(26,29);(7,9,11,10)(8,12,14,13)(15,17,19,25,18,24,26,23)(16,20,22,27,21,28,29,30);(3,4,5,6)(16,21)(17,23)(19,26)(20,27)(24,25)(28,30);(7,10,11,9)(8,13,14,12)(17,24)(20,28)(23,25)(27,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6583220
maximal normal subgroup ofC2×#6583220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373430);