Finite Group Databasebeta

← Prev (#353179) Next (#353181) →

Cayley diagram

C42×#99 #353180

order512
nilpotency class3
isoclinismisoclinic class of #99
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C24×C43

Aliases

direct productC42×#99

Subgroups

subgroupG/H
centerC2×C42C22⋊C4
commutatorC22C2×C43
Frattini, cosocleC25C24
socleC23C22×C22⋊C4
prime radicalC23×D4C23
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

2754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D4×#99, C42×B(4,𝔽2), C4×#25771, …
maximal normal subgroup ofC4×D4×#99, C42×B(4,𝔽2), C4×#25771, …
maximal quotient of#72312339, #72312728, #97×#99

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260376);