#503088

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

699 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=ei,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,g^c=gi

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, 410284);