#499987

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61258	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

918 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)(4,6)(5,8)(9,10)(11,21)(12,18)(13,30)(14,32)(15,20)(16,23)(17,26)(19,28)(22,35)(24,40)(25,39)(27,38)(29,37)(31,36)(33,34);(1,3,4,8)(2,5,6,7)(11,22)(12,26)(13,29)(14,28)(16,34)(17,18)(19,32)(21,35)(23,33)(24,31)(25,27)(30,37)(36,40)(38,39);(9,11,15,25)(10,16,20,36)(12,27,29,22)(13,31,26,23)(14,33,28,24)(17,34,30,40)(18,21,37,39)(19,38,32,35);(2,6)(5,7)

Polycyclic presentation:

2^9|b^2=e,c^2=i,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, 407183);