Finite Group Databasebeta

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Cayley diagram

C2×#10774 #353355

order512
nilpotency class3
isoclinismisoclinic class of #471
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C29

Aliases

direct productC2×#10774

Subgroups

subgroupG/H
center, socleC24#121
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

4250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72215768, #72215990, C2×#378855
maximal normal subgroup of#72215768, #72215990, C2×#378855
maximal quotient ofC2×#106211

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260551);