#6575157

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2154 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);(6,9)(10,12)(13,14,18,22)(15,23,27,34)(16,28,30,38)(17,31,32,21)(19,24,36,26)(20,33,37,29)(25,35,41,44)(39,42,43,40);(2,4)(13,15,18,27)(14,19,22,36)(16,25,30,41)(17,24,32,26)(20,35,37,44)(21,34,31,23)(28,39,38,43)(29,42,33,40);(5,7,8,11)(6,10,9,12)(15,24)(16,29)(17,32)(19,34)(20,28)(21,31)(23,36)(25,41)(26,27)(30,33)(35,44)(37,38);(13,16,18,30)(14,20,22,37)(15,25,27,41)(17,33,32,29)(19,35,36,44)(21,38,31,28)(23,39,34,43)(24,40,26,42)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,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, 6482353);