Finite Group Databasebeta

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Cayley diagram

C2×#25752 #6443309

order512
nilpotency class3
isoclinismisoclinic class of #24468
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25752

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341471, #72341496, C2×#6443319
maximal normal subgroup of#72341471, #72341496, C2×#6443319

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350505);