#500927

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=e,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,d^c=di,f^c=fh,g^c=ghi,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, 408123);