#144635

order	512
nilpotency class	4
isoclinism	isoclinic class of #144635
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10390
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,f^c=fi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72268762, #72268763

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51831);