Finite Group Databasebeta

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Cayley diagram

C2×#10567 #241123

order512
nilpotency class3
isoclinismisoclinic class of #9914
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#10567

Subgroups

subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1684 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);(3,4,7,9)(5,10,6,8)(11,12)(13,15)(14,17)(16,18)(19,20)(21,23)(22,24)(25,26);(3,5,7,6)(4,8,9,10)(19,21)(20,23)(22,25)(24,26);(3,6,7,5)(4,8,9,10)(12,15)(13,18)(16,17)(23,26)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267852, #72267857
maximal normal subgroup of#72267852, #72267857
maximal quotient of#72267857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148319);