Finite Group Databasebeta

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Cayley diagram

C2×#10827 #353547

order512
nilpotency class3
isoclinismisoclinic class of #471
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#10827

Subgroups

subgroupG/H
center, socleC24#121
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1970 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)(12,17)(13,21)(14,15)(18,25)(19,20)(24,26);(7,8,9,10)(11,12,15,19,16,20,14,17)(13,18,23,25,21,26,22,24);(11,13,14,22,16,21,15,23)(12,18,17,24,20,26,19,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72215768, #72216072, C2×#378855
maximal normal subgroup of#72215768, #72216072, C2×#378855
maximal quotient ofC2×#108656

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260743);