Finite Group Databasebeta

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Cayley diagram

C2×#7857 #180865

order512
nilpotency class2
isoclinismisoclinic class of #7855
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42

Aliases

direct productC2×#7857

Subgroups

subgroupG/H
centerC22×C4C2×C42
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,11,13,17)(6,9)(7,10)(8,12,15,18)(14,16)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(3,5)(4,8,10,15)(6,12)(7,13)(9,11,16,17)(14,18)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(21,27)(24,30)(28,33)(31,34)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303344, C2×#233574, #72303447
maximal normal subgroup of#72303344, C2×#233574, #72303447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88061);