#500348

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 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)(3,8)(4,6)(5,7)(9,10,16,23)(11,17,29,45)(12,30,36,66)(13,19,38,22)(14,21,37,51)(15,20,40,52)(18,46,50,33)(24,56,59,72)(25,42,28,54)(26,60,39,44)(27,53,62,43)(31,65,61,69)(32,49,35,68)(34,55,67,48)(41,63,70,58)(47,64,71,57);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,31)(13,25)(14,27)(15,39)(16,29)(18,47)(19,42)(20,44)(21,53)(22,54)(23,45)(24,34)(26,40)(28,38)(30,65)(32,58)(33,57)(35,63)(36,61)(37,62)(41,49)(43,51)(46,64)(48,72)(50,71)(52,60)(55,56)(59,67)(66,69)(68,70);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,24)(13,32)(14,34)(15,33)(16,36)(17,41)(19,48)(20,49)(21,30)(22,55)(23,50)(25,57)(26,61)(27,63)(28,64)(29,59)(31,39)(35,38)(37,67)(40,46)(42,69)(43,71)(44,72)(45,70)(47,53)(51,66)(52,68)(54,65)(56,60)(58,62);(2,7)(6,8)(9,13,16,38)(10,19,23,22)(11,25,29,28)(12,32,36,35)(14,15,37,40)(17,42,45,54)(18,48,50,55)(20,51,52,21)(24,57,59,64)(26,27,39,62)(30,49,66,68)(31,58,61,63)(33,67,46,34)(41,69,70,65)(43,60,53,44)(47,72,71,56)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407544);