Finite Group Databasebeta

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Cayley diagram

C2×#21147 #6371851

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#21147

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6380033, C2×#6381822, C2×#6372769
maximal normal subgroup ofC2×#6380033, C2×#6381822, C2×#6372769
maximal quotient ofC2×#203677

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279047);