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Cayley diagram

C2×#12677 #493157

order512
nilpotency class4
isoclinismisoclinic class of #2180
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#12677

Subgroups

subgroupG/H
center, socleC23#442
commutatorC8C22×C42
Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#495949, C2×#496007, …
maximal normal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#495949, C2×#496007, …
maximal quotient ofC2×#129165, C2×#129120, C2×#129171, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400353);