Finite Group Databasebeta

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Cayley diagram

C2×#23223 #6382865

order512
nilpotency class3
isoclinismisoclinic class of #3264
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#23223

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2706 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);(3,4,6,5)(8,12)(13,14)(15,16)(17,20)(18,22)(19,21);(3,5,6,4)(7,8)(9,13)(10,14)(11,12)(15,17,19,22)(16,20,21,18);(7,9,11,10)(8,13,12,14)(15,18,22,16,19,20,17,21);(7,10,11,9)(8,13,12,14)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6384086, C2×#6624966, C2×#6620116
maximal normal subgroup ofC2×#6384086, C2×#6624966, C2×#6620116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290061);