C₂×#7541 #153563

order	512
nilpotency class	6
isoclinism	isoclinic class of #7540
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7541

Subgroups

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂²×D₃₂	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(1,2);(3,4,6,13,10,16,9,34)(5,17,20,44,24,11,23,35)(7,27,25,60,31,40,30,62)(8,32,26,48,33,63,29,15)(12,38,41,18,43,51,64,21)(14,46,45,19,39,54,61,28)(22,57,52,37,59,47,55,50)(36,65,42,56,49,58,66,53);(4,11)(5,18)(6,9)(7,28)(8,29)(12,39)(13,44)(14,43)(15,47)(16,17)(19,31)(20,38)(21,24)(22,58)(23,51)(25,54)(26,33)(27,36)(30,46)(32,37)(34,35)(40,49)(41,45)(42,62)(48,57)(50,63)(52,56)(53,55)(59,65)(60,66)(61,64)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#153598, D₃×#7541, #72286491, …
maximal normal subgroup of	C₂×#153598, #72286491, #72286831, …
maximal quotient of	D₃×#7541, #72286879

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60759);