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Cayley diagram

C2×#13136 #494869

order512
nilpotency class4
isoclinismisoclinic class of #2212
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#13136

Subgroups

subgroupG/H
center, socleC23#459
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

912 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);(3,4,6,5)(8,11)(9,17)(10,12)(14,19)(15,16)(18,20)(21,22);(3,5,6,4)(7,8,12,15,13,16,10,11)(9,14,21,22,19,17,18,20);(7,9,10,18,13,19,12,21)(8,14,11,20,16,17,15,22)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×#2213, #72242481
maximal normal subgroup ofC2×C4×#2213, #72242481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402065);