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Cayley diagram

C16×#120 #495667

order512
nilpotency class2
isoclinismisoclinic class of #120
automorphism?
homologyAbelianization (H1) = C23×C16 · Schur multiplier (H2) = C27

Aliases

direct productC16×#120

Subgroups

subgroupG/H
centerC22×C16C23
commutatorC22C23×C16
Frattini, cosocleC22×C8C24
socleC23C23×C8
prime radicalC2×#120C8
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2)(3,4)(5,6)(7,8);(2,4);(6,8)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#120×AGL(1,𝔽17), C16×B(4,𝔽2), C16×#1020, …
maximal normal subgroup ofC16×B(4,𝔽2), #72287670, #72319372, …
maximal quotient of#120×AGL(1,𝔽17), C16×B(4,𝔽2), C16×#453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402863);