Finite Group Databasebeta

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Cayley diagram

C2×#23945 #6451685

order512
nilpotency class3
isoclinismisoclinic class of #23924
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23945

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(7,8)(9,13)(10,12)(11,14)(15,16)(17,24)(18,22)(19,21)(20,25)(23,27)(26,29)(28,30);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(15,17)(16,20)(18,25)(19,26)(21,28)(22,24)(23,29)(27,30);(4,6)(16,21)(19,27)(20,28)(22,23)(24,29)(26,30);(8,12)(11,13)(16,22)(17,25)(21,23)(26,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264792, #72264809, #72264841
maximal normal subgroup of#72264792, #72264809, #72264841

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358881);