#499986

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

982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,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×#499986
maximal normal subgroup of	#72319933, C₂×#499986
maximal quotient of	C₂×#499986

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407182);