Finite Group Databasebeta

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Cayley diagram

C2×#12805 #493731

order512
nilpotency class4
isoclinismisoclinic class of #11521
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#12805

Subgroups

subgroupG/H
centerC2×C4#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×#41C24
socleC22#1799
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

1176 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)(7,20,21,10)(8,12)(9,14)(11,19)(15,29,30,25)(16,26)(18,24,32,31)(22,27,33,34)(23,28);(3,5,6,14,8,16,19,28)(4,9,11,13,12,23,17,26)(7,15,18,27,21,30,32,34)(10,22,24,25,20,33,31,29);(4,10)(9,22)(11,24)(12,20)(13,25)(17,31)(23,33)(26,29)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#130854, #72250315, #72250320, …
maximal normal subgroup ofC2×#130854, #72250315, #72250320, …
maximal quotient of#72264399, #72264419, #72308038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400927);