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Cayley diagram

C4⋊C4×#99 #356182

order512
nilpotency class3
isoclinismisoclinic class of #17411
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC4⋊C4×#99

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

2150 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:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D4×#99, C4⋊C4×B(4,𝔽2), #72311736, …
maximal normal subgroup ofC4×D4×#99, C4⋊C4×B(4,𝔽2), #72311736, …
maximal quotient of#72312156, #72312540, #99×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263378);