Finite Group Databasebeta

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Cayley diagram

C2×#24085 #6452946

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24085

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6453878, C2×#6453870, C2×#6452952
maximal normal subgroup ofC2×#6453878, C2×#6453870, C2×#6452952
maximal quotient ofC2×#257044, #72309281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360142);