Finite Group Databasebeta

← Prev (#7626175) Next (#7626177) →

Cayley diagram

C4×#3558 #7626176

order512
nilpotency class2
isoclinismisoclinic class of #3532
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC4×#3558

Subgroups

subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC2×#3558C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

7886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72297207, C2×C4×#3558, #72304030
maximal normal subgroup of#72297207, C2×C4×#3558, #72304030
maximal quotient of#72297305, #72297472, C4×#44877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533372);