Finite Group Databasebeta

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Cayley diagram

C2×#22264 #6376396

order512
nilpotency class2
isoclinismisoclinic class of #2613
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#22264

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC22×#127C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6376471, #72264549, #72264767
maximal normal subgroup ofC2×#6376471, #72264549, #72264767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283592);