Finite Group Databasebeta

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Cayley diagram

C2×#9951 #235040

order512
nilpotency class3
isoclinismisoclinic class of #9946
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#9951

Subgroups

subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258158
maximal normal subgroup of#72258158

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142236);