#6676074

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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)(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,36)(26,27)(28,38)(29,39)(30,35)(31,33)(32,40)(34,37)(41,43)(42,44);(5,6)(7,10)(8,12)(9,11)(13,15,19,28)(14,20,23,38)(16,29,32,41)(17,26,24,37)(18,34,22,27)(21,39,40,43)(25,35,42,31)(30,44,33,36);(2,4)(13,16,19,32)(14,21,23,40)(15,25,28,42)(17,33,24,30)(18,35,22,31)(20,36,38,44)(26,43,37,39)(27,41,34,29);(5,7,9,8)(6,10,11,12)(13,17,19,24)(14,22,23,18)(15,26,28,37)(16,30,32,33)(20,27,38,34)(21,35,40,31)(25,39,42,43)(29,44,41,36);(5,8,9,7)(6,10,11,12)(14,23)(15,27)(18,22)(20,37)(21,40)(25,41)(26,38)(28,34)(29,42)(31,35)(36,39)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6583270);