Finite Group Databasebeta

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Cayley diagram

C2×#17808 #6344332

order512
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#17808

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C22×C4⋊C4
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#121058, #72240049, #72240051
maximal normal subgroup ofC2×#121058, #72240049, #72240051
maximal quotient of#72240046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251528);