#500144

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
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	#61051	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

974 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)(12,30)(13,19)(14,39)(15,20)(16,23)(18,33)(21,37)(22,38)(24,57)(25,42)(26,51)(27,43)(28,55)(29,45)(31,65)(32,52)(34,56)(35,48)(36,54)(40,50)(41,60)(44,63)(46,59)(47,58)(49,68)(53,62)(61,72)(64,71)(66,70)(67,69);(2,5)(6,8)(9,11,16,29)(10,17,23,45)(12,31,36,67)(13,25,38,28)(14,27,37,62)(15,26,40,63)(18,46,49,71)(19,42,22,55)(20,51,50,44)(21,53,39,43)(24,58,61,34)(30,65,54,69)(32,66,35,60)(33,59,68,64)(41,52,70,48)(47,72,56,57);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,32)(14,34)(15,33)(16,36)(17,41)(19,47)(20,52)(21,54)(22,56)(23,49)(25,59)(26,31)(27,60)(28,64)(29,61)(30,39)(35,38)(37,58)(40,68)(42,65)(43,71)(44,72)(45,70)(46,53)(48,50)(51,57)(55,69)(62,66)(63,67);(2,5)(6,8)(9,13,16,38)(10,19,23,22)(11,25,29,28)(12,32,36,35)(14,15,37,40)(17,42,45,55)(18,47,49,56)(20,21,50,39)(24,59,61,64)(26,62,63,27)(30,52,54,48)(31,66,67,60)(33,58,68,34)(41,65,70,69)(43,51,53,44)(46,72,71,57)

Polycyclic presentation:

2^9|b^2=i,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=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407340);