Finite Group Databasebeta

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Cayley diagram

C2×#9179 #197219

order512
nilpotency class3
isoclinismisoclinic class of #9159
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9179

Subgroups

subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1782 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);(3,4,5,8,6,9,10,7)(11,12,15,18,16,20,14,19)(13,21,23,25,24,26,22,17);(4,7)(5,10)(8,9)(11,13)(12,17,20,25)(14,22)(15,23)(16,24)(18,21,19,26);(12,18)(14,15)(17,21)(19,20)(22,23)(25,26)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72278842, C2×#197248
maximal normal subgroup of#72278842, C2×#197248

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104415);