Finite Group Databasebeta

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Cayley diagram

C2×#9387 #200436

order512
nilpotency class3
isoclinismisoclinic class of #9383
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#9387

Subgroups

subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#392C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,6,9,7,10,5,8)(11,12)(13,20,21,24)(14,17,22,23)(15,18)(16,19)(25,26);(4,8,10,9)(11,13,15,23,16,21,25,17)(12,14,18,24,19,22,26,20);(11,14)(12,17)(13,18)(15,24)(16,22)(19,23)(20,25)(21,26)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#130056, #72340417, #72340419
maximal normal subgroup ofC2×#130056, #72340417, #72340419

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107632);