#6575631

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,30)(24,41)(25,35)(26,37)(28,44)(29,40)(31,36)(33,39)(34,43)(38,42);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,23,37)(16,28,29,24)(17,31,18,25)(21,38,39,34)(22,36,32,35)(27,44,40,41)(30,42,33,43);(2,4)(13,16,19,29)(14,21,23,39)(15,24,26,28)(17,30,18,33)(20,34,37,38)(22,40,32,27)(25,42,31,43)(35,41,36,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(16,29)(21,39)(24,28)(27,40)(30,33)(34,38)(41,44)(42,43);(2,4)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,30,29,33)(20,35,37,36)(21,40,39,27)(24,42,28,43)(34,41,38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482827);