Finite Group Databasebeta

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Cayley diagram

C2×#12962 #494359

order512
nilpotency class4
isoclinismisoclinic class of #2196
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#12962

Subgroups

subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×D4C23×C4
Frattini, cosocleC2×#41C24
socleC22#1867
prime radicalC2×#525C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

1127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303731, C2×#494361, C2×#512038, …
maximal normal subgroup of#72303731, C2×#494361, C2×#512038, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401555);