#6575167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482363);