Finite Group Databasebeta

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Cayley diagram

C2×#13395 #499046

order512
nilpotency class4
isoclinismisoclinic class of #13309
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13395

Subgroups

subgroupG/H
centerC2×C4#469
commutator, Frattini, cosocleC22×C8C24
socleC22#2034
prime radicalC2×#3050C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1146 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)(5,13)(6,17)(7,22)(8,11)(9,15)(10,20)(12,19)(14,29)(16,26)(18,25)(21,33)(23,30)(24,31)(27,34)(28,32);(9,24)(13,29)(25,28)(26,30);(3,5,7,23,8,14,19,16)(4,9,12,28,11,24,22,25)(6,18,20,31,21,32,34,15)(10,26,17,29,27,30,33,13)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72231484, #72231488, #72231523, …
maximal normal subgroup of#72231484, #72231488, #72231523, …
maximal quotient of#72252902, #72252908, #72257093, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406242);