#508021

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,27)(13,19)(14,34)(15,38)(16,21)(18,30)(20,36)(22,50)(23,40)(24,54)(25,42)(26,43)(28,59)(29,61)(31,48)(32,62)(33,49)(35,45)(37,60)(39,51)(41,56)(44,55)(46,63)(47,64)(52,71)(53,70)(57,69)(58,68)(65,67)(66,72);(2,5)(6,8)(9,11)(10,17)(12,28)(13,23)(14,24)(15,25)(16,26)(18,44)(19,40)(20,41)(21,43)(22,31)(27,59)(29,52)(30,55)(32,58)(33,53)(34,54)(35,65)(36,56)(37,57)(38,42)(39,46)(45,67)(47,66)(48,50)(49,70)(51,63)(60,69)(61,71)(62,68)(64,72);(1,3)(2,6)(4,7)(5,8)(9,12,16,32)(10,18,21,47)(11,22,26,52)(13,29,15,31)(14,35,37,63)(17,39,43,67)(19,45,38,46)(20,48,49,61)(23,28,25,58)(24,55,57,72)(27,60,62,34)(30,33,64,36)(40,44,42,66)(41,68,70,59)(50,69,71,54)(51,53,65,56);(3,7)(6,8)(9,13,16,15)(10,19,21,38)(11,23,26,25)(12,29,32,31)(14,33,37,36)(17,40,43,42)(18,45,47,46)(20,34,49,60)(22,28,52,58)(24,53,57,56)(27,61,62,48)(30,35,64,63)(39,44,67,66)(41,54,70,69)(50,59,71,68)(51,55,65,72)

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415217);