Finite Group Databasebeta

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Cayley diagram

C2×#12778 #493617

order512
nilpotency class4
isoclinismisoclinic class of #12778
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#12778

Subgroups

subgroupG/H
center, socleC22#1790
commutatorC2×Q8C23×C4
Frattini, cosocleC2×#41C24
prime radicalC2×#3549C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

2781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72219502, C2×#511819, C2×#11714278, …
maximal normal subgroup of#72219502, C2×#511819, #72236900, …
maximal quotient of#72219495, #72231007, #72231286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400813);