#512236

order	512
nilpotency class	4
isoclinism	isoclinic class of #511896
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,24,25,33)(13,27)(14,16)(15,20)(17,23,34,39)(18,29)(19,22)(26,37,28,38)(30,32)(31,36,40,35);(1,3,4,7)(2,5,6,8)(11,22)(14,30)(16,32)(19,21)(23,39)(24,33)(31,40)(37,38);(2,6)(5,8)(9,11,13,14,15,22,29,30)(10,16,18,21,20,32,27,19)(12,23,28,31,25,39,26,40)(17,33,36,38,34,24,35,37);(9,12)(10,17)(11,23)(13,28)(14,31)(15,25)(16,33)(18,36)(19,37)(20,34)(21,38)(22,39)(24,32)(26,29)(27,35)(30,40)

Polycyclic presentation:

2^9|a^2=e,b^2=fhi,c^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419432);