Finite Group Databasebeta

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Cayley diagram

C2×#32942 #6629025

order512
nilpotency class3
isoclinismisoclinic class of #32941
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#32942

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4730 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);(3,4)(5,6)(8,12)(9,16)(10,13)(15,20)(17,18)(19,21);(7,8,13,20,11,15,10,12)(9,14,16,21,18,22,17,19);(7,9)(8,14)(10,17)(11,18)(12,19)(13,16)(15,22)(20,21);(4,6)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72322763, #72322788, #72322798, …
maximal normal subgroup of#72322763, #72322788, #72322798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536221);