Finite Group Databasebeta

← Prev (#358567) Next (#358569) →

Cayley diagram

C2×#11367 #358568

order512
nilpotency class3
isoclinismisoclinic class of #2031
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11367

Subgroups

subgroupG/H
centerC22×C4#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23#400
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268632, #72268633, #72268635
maximal normal subgroup of#72268632, #72268633, #72268635

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265764);