#6487822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 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)(10,15)(11,19)(13,28)(17,30)(18,31)(21,35)(22,23)(24,34)(25,26)(29,33)(37,38)(39,40);(3,4)(6,7)(9,10,13,15,14,18,28,31)(11,20,22,33,23,16,19,29)(12,24,26,34,27,17,25,30)(21,32,37,39,38,36,35,40);(1,3,5,4)(2,6,8,7)(9,11,14,23)(10,16,18,20)(12,21,27,38)(13,22,28,19)(15,29,31,33)(17,32,24,36)(25,35,26,37)(30,39,34,40);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,21)(13,26)(14,27)(15,30)(16,32)(18,24)(19,35)(20,36)(22,37)(23,38)(25,28)(29,39)(31,34)(33,40);(2,8)(6,7)(9,13,14,28)(10,15,18,31)(11,22,23,19)(12,25,27,26)(16,29,20,33)(17,34,24,30)(21,35,38,37)(32,40,36,39)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6395018);