Finite Group Databasebeta

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Cayley diagram

C2×#17638 #6343743

order512
nilpotency class3
isoclinismisoclinic class of #327
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#17638

Subgroups

subgroupG/H
centerC2×C42C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C22×C22⋊C4
prime radicalC2×#3444C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

3764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(4,8,9,10)(11,12,15,18,16,13,14,17);(3,4)(5,8)(6,10)(7,9)(11,13)(12,16)(14,18)(15,17);(3,5,7,6)(4,8,9,10)(11,14,16,15)(12,17,13,18);(11,14,16,15)(12,17,13,18)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209967, #72209971, C2×#6343786, …
maximal normal subgroup of#72209967, #72209971, C2×#6343786, …
maximal quotient of#72239496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250939);