Finite Group Databasebeta

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Cayley diagram

C2×#25924 #6447717

order512
nilpotency class3
isoclinismisoclinic class of #24464
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25924

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3038C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3674 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);(3,4,7,10)(5,9,6,8)(11,12)(13,16)(14,17)(15,18);(3,5,7,6)(4,8,10,9)(11,13,16,12)(14,15,18,17);(3,5,7,6)(4,9,10,8)(11,14,15,13,16,18,17,12);(3,6,7,5)(4,9,10,8)(11,15,16,17)(12,14,13,18)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271418, #72271451
maximal normal subgroup of#72271418, #72271451

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354913);