Finite Group Databasebeta

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Cayley diagram

C2×#13029 #494541

order512
nilpotency class4
isoclinismisoclinic class of #2212
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#13029

Subgroups

subgroupG/H
center, socleC23#459
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#509C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72242475, C2×C4×#2213
maximal normal subgroup of#72242475, C2×C4×#2213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401737);