Finite Group Databasebeta

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Cayley diagram

C2×#32923 #6628965

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

Aliases

direct productC2×#32923

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4566 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)(7,8,12,10)(9,15,18,22)(11,19,21,14)(13,20,16,17);(3,4,5,6);(7,9)(8,13)(10,16)(11,20)(12,18)(14,22)(15,19)(17,21);(7,10,12,8)(9,16,18,13)(11,14,21,19)(15,20,22,17)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271441, #72271446, #72271558
maximal normal subgroup of#72271441, #72271446, #72271558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536161);