#6575531

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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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)(15,23,26,38)(16,27,29,36)(17,30,32,21)(19,31,35,24)(20,28,37,33)(25,41,43,34)(39,42,44,40);(2,4)(13,15)(14,19)(16,25)(17,31)(18,26)(20,34)(21,38)(22,35)(23,30)(24,32)(27,44)(28,40)(29,43)(33,42)(36,39)(37,41);(5,7,8,11)(6,10,9,12)(15,24)(16,28)(17,32)(19,23)(20,36)(21,30)(26,31)(27,37)(29,33)(35,38)(39,44)(40,42);(13,16,18,29)(14,20,22,37)(15,25,26,43)(17,33,32,28)(19,34,35,41)(21,27,30,36)(23,39,38,44)(24,40,31,42)

Polycyclic presentation:

2^9|b^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6482727);