Finite Group Databasebeta

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Cayley diagram

C2×#23982 #6451803

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23982

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4870 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)(7,8)(9,13)(10,12)(11,14)(15,16,20,17)(18,22,19,21);(7,9,10,14)(8,11,12,13)(15,17,20,16)(18,21,19,22);(4,6)(16,21)(17,22)(18,19);(8,12)(11,13)(15,18,20,19)(16,22,17,21)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264772, #72264809, #72264824, …
maximal normal subgroup of#72264772, #72264809, #72264824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358999);