Finite Group Databasebeta

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Cayley diagram

C2×#7992 #181278

order512
nilpotency class3
isoclinismisoclinic class of #1284
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26

Aliases

direct productC2×#7992

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1746 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)(5,9)(6,8)(7,10)(11,12,14,18,16,19,15,24)(13,20,22,17,23,25,21,26);(4,8,10,9)(11,13,15,22,16,23,14,21)(12,17,18,20,19,26,24,25);(3,5,7,6)(4,9,10,8)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239498, #72239500, #72239501
maximal normal subgroup of#72239498, #72239500, #72239501
maximal quotient of#72239505

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88474);