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Cayley diagram

C2×#7155 #126650

order512
nilpotency class4
isoclinismisoclinic class of D16
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×#7155

Subgroups

subgroupG/H
centerC2×C42D8
commutatorC8C2×C4×C8
Frattini, cosocleC2×C4×C8C23
socleC23#340
prime radicalC24#106
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,22)(14,27)(15,30)(16,32)(17,21)(18,23)(19,29)(20,28)(24,34)(25,42)(26,38)(31,35)(33,41)(36,40)(37,39);(11,13,15,24,28,39,29,40,17,26,14,23,16,25,31,41)(12,18,19,22,32,36,30,42,21,34,35,38,20,33,27,37)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#129096, C2×C8×#359, C2×#128669
maximal normal subgroup ofC2×#129096, C2×C8×#359, C2×#128669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33846);