Finite Group Databasebeta

← Prev (#10574232) Next (#10574234) →

Cayley diagram

C22×#2971 #10574233

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC22×#2971

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C25×C4
Frattini, cosocleC2×C4C26
socleC24C23×C4
prime radicalC23×#41C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

8262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#61465, C22×#60283
maximal normal subgroup ofC22×#61465, C22×#60283

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481429);