#500584

order	512
nilpotency class	4
isoclinism	isoclinic class of #499919
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	#60682	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

799 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)(3,7)(4,5)(6,8)(9,10)(11,23)(12,17)(13,22)(14,20)(15,30)(16,19)(18,27)(21,32)(24,34)(25,37)(26,38)(28,39)(29,33)(31,35)(36,40);(2,5)(6,7)(9,11)(10,17)(12,14)(13,26)(15,27)(16,24)(18,33)(19,38)(20,23)(21,36)(22,34)(25,29)(28,40)(30,37)(31,32)(35,39);(9,12,14,11)(10,18,20,37)(13,26,16,24)(15,31,29,40)(17,33,23,30)(19,36,22,35)(21,34,39,38)(25,32,27,28);(1,3,4,8)(2,6,5,7)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,34,23,38)(18,36,37,35)(21,33,39,30)(25,40,27,31)

Polycyclic presentation:

2^9|c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,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, 407780);