C₂×#7540 #153561

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

Aliases

direct product

C₂×#7540

Subgroups

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂×#2262	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

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

normal subgroups only

This group has 186 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);(4,10,11,32)(5,15)(6,21)(7,23)(8,27)(12,38,36,46)(13,40,37,45)(14,41,29,35)(16,22,33,31)(17,25)(18,50)(19,54)(20,30)(24,49)(26,34,62,64)(28,60)(39,51,58,66)(42,43)(44,52)(47,61,55,59)(48,63)(53,65)(56,57);(3,4)(5,16)(6,22)(7,26)(8,29)(9,11)(10,30)(12,21)(13,23)(14,27)(15,32)(17,36)(18,51)(19,55)(20,33)(24,61)(25,31)(28,62)(34,65)(35,44)(37,60)(38,63)(39,49)(40,48)(41,54)(42,46)(43,45)(47,52)(50,64)(53,66)(56,59)(57,58)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#153598, D₃×#7540, #72286831, …
maximal normal subgroup of	C₂×#153598, #72286831, #72286857, …
maximal quotient of	D₃×#7540, #72286857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60757);