#500335

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	#61258	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

760 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,18)(13,17)(14,31)(15,32)(16,23)(19,28)(20,29)(22,36)(24,40)(25,35)(26,37)(27,38)(30,39)(33,34);(1,3,5,4)(2,6,7,8)(9,11,12,23)(10,16,18,21)(13,30,27,25)(14,22,28,24)(15,33,29,26)(17,35,38,39)(19,36,31,40)(20,34,32,37);(1,4,5,3)(2,6,7,8)(10,17)(11,22)(13,28)(14,27)(15,29)(16,34)(18,38)(19,20)(21,37)(23,24)(25,33)(26,30)(31,32)(36,40);(2,7)(6,8)(10,18)(16,21)(17,38)(19,31)(20,32)(34,37)(35,39)(36,40)

Polycyclic presentation:

2^9|b^2=e,c^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,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319933, C₂×#500335
maximal normal subgroup of	#72319933, C₂×#500335
maximal quotient of	C₂×#500335

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407531);