#6487510

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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	#62848	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3952 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)(9,10,16,12)(11,20,24,32)(13,18,28,25)(14,19,15,26)(17,30,21,23)(22,35,39,40)(27,33,29,34)(31,37,36,38);(3,4)(6,7)(9,11)(10,17)(12,21)(13,22)(14,23)(15,30)(16,24)(18,31)(19,32)(20,26)(25,36)(27,37)(28,39)(29,38)(33,35)(34,40);(1,3,5,4)(2,6,8,7)(9,12,16,10)(11,21,24,17)(13,25,28,18)(14,26,15,19)(20,30,32,23)(22,36,39,31)(27,34,29,33)(35,37,40,38);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,22)(12,25)(14,29)(15,27)(16,28)(17,31)(19,34)(20,35)(21,36)(23,38)(24,39)(26,33)(30,37)(32,40);(2,8)(6,7)(9,14,16,15)(10,19,12,26)(11,23,24,30)(13,27,28,29)(17,32,21,20)(18,33,25,34)(22,37,39,38)(31,35,36,40)

Polycyclic presentation:

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

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, 6394706);