Finite Group Databasebeta

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Cayley diagram

C2×#32921 #6628964

order512
nilpotency class3
isoclinismisoclinic class of #32888
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#32921

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

4782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271414, #72271418, #72271448, …
maximal normal subgroup of#72271414, #72271418, #72271448, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536160);