Finite Group Databasebeta

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Cayley diagram

C2×#28932 #6459970

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#28932

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6742229, C2×#6460700, C2×#6699142
maximal normal subgroup ofC2×#6742229, C2×#6460700, C2×#6699142

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367166);