Finite Group Databasebeta

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Cayley diagram

C22×#3375 #10576465

order512
nilpotency class3
isoclinismisoclinic class of #3264
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC22×#3375

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC22×#127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

6950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#20150
maximal normal subgroup ofC22×#20150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10483661);