#6670202

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(6,9)(7,11)(13,14,18,22)(15,19,25,34)(16,20,27,36)(17,21,31,40)(23,32,41,43)(24,33,29,38)(26,35,42,44)(28,37,30,39);(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,19,22,34)(16,26,27,42)(17,29,31,24)(20,35,36,44)(21,38,40,33)(23,30,41,28)(32,39,43,37);(2,4)(13,16)(14,20)(15,23)(17,30)(18,27)(19,32)(21,39)(22,36)(24,42)(25,41)(26,29)(28,31)(33,44)(34,43)(35,38)(37,40);(5,7,8,11)(6,10,9,12)(16,27)(20,36)(23,41)(26,42)(28,30)(32,43)(35,44)(37,39);(6,9)(10,12)(15,24)(17,31)(19,33)(21,40)(23,42)(25,29)(26,41)(28,30)(32,44)(34,38)(35,43)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,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, 6577398);