#500923

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	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

703 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)(18,30)(20,36)(21,35)(23,57)(24,50)(25,42)(26,55)(27,45)(29,65)(31,56)(32,47)(33,54)(37,52)(39,44)(40,51)(41,59)(43,62)(46,60)(48,58)(49,68)(53,63)(61,72)(64,71)(66,70)(67,69);(1,3,5,4)(2,6,8,7)(9,11,16,27)(10,17,22,45)(12,29,33,67)(13,25,36,63)(14,24,37,62)(15,39,35,26)(18,46,49,71)(19,50,52,43)(20,53,34,42)(21,55,38,44)(23,58,61,31)(28,65,54,69)(30,60,68,64)(32,59,40,66)(41,51,70,47)(48,72,56,57);(9,12,16,33)(10,18,22,49)(11,23,27,61)(13,31,36,58)(14,30,37,68)(15,40,35,32)(17,41,45,70)(19,51,52,47)(20,54,34,28)(21,56,38,48)(24,29,62,67)(25,59,63,66)(26,64,39,60)(42,71,53,46)(43,72,50,57)(44,65,55,69);(1,4,5,3)(2,7,8,6)

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,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, 408119);