#6487501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi

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, 6394697);