#6574649

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 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)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11)(13,14)(15,24)(16,21)(17,31)(18,34)(19,23)(20,26)(22,36)(25,39)(27,37)(28,43)(29,42)(30,41)(32,44)(33,38)(35,40);(2,4)(13,15,19,17)(14,20,23,38)(16,27,30,28)(18,33,36,26)(21,39,41,44)(22,31,34,24)(25,29,32,35)(37,40,43,42);(13,16,19,30)(14,21,23,41)(15,25,17,32)(18,35,36,29)(20,37,38,43)(22,42,34,40)(24,39,31,44)(26,27,33,28);(13,17,19,15)(14,20,23,38)(16,28,30,27)(18,26,36,33)(21,39,41,44)(22,31,34,24)(25,35,32,29)(37,40,43,42)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,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, 6481845);