#6574521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

1899 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)(16,27)(21,28)(25,33)(29,30)(31,36)(38,44)(40,43)(41,42);(5,7,8,12)(6,9,10,11)(13,14,19,23)(15,24,17,32)(16,28,27,21)(18,35,37,22)(20,34,39,26)(25,43,33,40)(29,44,30,38)(31,41,36,42);(2,4)(13,15,19,17)(14,20,23,39)(16,29,27,30)(18,34,37,26)(21,40,28,43)(22,32,35,24)(25,31,33,36)(38,41,44,42);(13,16,19,27)(14,21,23,28)(15,25,17,33)(18,36,37,31)(20,38,39,44)(22,42,35,41)(24,40,32,43)(26,29,34,30);(13,17,19,15)(14,20,23,39)(16,30,27,29)(18,26,37,34)(21,40,28,43)(22,32,35,24)(25,36,33,31)(38,41,44,42)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481717);