#505656

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,16,20)(11,14,23,31)(12,24,29,40)(13,22,30,19)(15,18,34,32)(17,25,36,27)(21,38,35,33)(26,37,28,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,20,16,23,31,10)(12,17,28,35,29,36,26,21)(13,22,15,32,30,19,34,18)(24,38,37,27,40,33,39,25);(3,7)(5,8)(9,12)(10,17)(11,21)(13,25)(14,26)(15,33)(16,29)(18,24)(19,37)(20,36)(22,39)(23,35)(27,30)(28,31)(32,40)(34,38);(9,13,14,15,16,30,31,34)(10,18,11,22,20,32,23,19)(12,25,26,33,29,27,28,38)(17,24,21,39,36,40,35,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,g^c=gi,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, 412852);