Finite Group Databasebeta

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Cayley diagram

C2×#9361 #200020

order512
nilpotency class3
isoclinismisoclinic class of #9345
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9361

Subgroups

subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2336 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)(5,7,9,10)(6,8)(11,12,15,19,16,20,14,17)(13,21,23,18,24,25,22,26);(3,5)(4,7,8,10)(6,9)(12,17,20,19)(18,21,26,25);(11,13)(12,18)(14,22)(15,23)(16,24)(17,21)(19,25)(20,26)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72231648, C2×#200068
maximal normal subgroup of#72231648, C2×#200068

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107216);