#6671699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2147 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)(6,9)(7,11);(5,6,8,9)(7,10,11,12)(13,14)(15,24)(16,21)(17,29)(18,32)(19,23)(20,25)(22,30)(26,41)(27,34)(28,38)(31,37)(33,36)(35,43)(39,40)(42,44);(2,4)(13,15,19,27)(14,20,23,36)(16,25,18,33)(17,26,31,43)(21,34,32,24)(22,35,38,41)(28,40,30,42)(29,44,37,39);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,32)(15,25,27,33)(17,30,31,28)(20,34,36,24)(22,37,38,29)(26,42,43,40)(35,39,41,44);(6,9)(10,12)(13,17,19,31)(14,22,23,38)(15,26,27,43)(16,28,18,30)(20,35,36,41)(21,29,32,37)(24,39,34,44)(25,40,33,42)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578895);