Finite Group Databasebeta

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Cayley diagram

C2×#8066 #181652

order512
nilpotency class3
isoclinismisoclinic class of #1504
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26

Aliases

direct productC2×#8066

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1530 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);(3,4,7,10,8,11,6,16)(5,12,15,17,13,18,14,9)(19,20)(21,28)(22,26)(23,32)(24,27)(25,30)(29,34)(31,33);(3,5)(4,9,11,17)(6,14)(7,15)(8,13)(10,12,16,18)(19,21,23,29,24,30,22,31)(20,25,26,34,27,28,32,33);(5,13)(9,17)(12,18)(14,15)(19,22,24,23)(20,26,27,32)(21,29,30,31)(25,33,28,34)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240034, C2×#121058, #72240053
maximal normal subgroup of#72240034, C2×#121058, #72240053
maximal quotient of#72240035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88848);