Finite Group Databasebeta

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Cayley diagram

C2×#20411 #6369844

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#20411

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC4C23×C42
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC23×#41C22
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5586 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)(7,8,10,9)(11,12)(13,21)(14,18)(15,19)(16,20)(17,23)(22,26)(24,25);(4,6)(11,13,15,24,16,22,14,23)(12,17,18,26,20,25,19,21);(7,9,10,8)(13,22)(17,25)(21,26)(23,24);(11,14,16,15)(12,18,20,19)(13,23,22,24)(17,26,25,21)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264892, #72264900, #72264927, …
maximal normal subgroup of#72264892, #72264900, #72264927, …
maximal quotient ofC2×#354203, #72308536, #72318799, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277040);