#6690871

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,23,28)(19,24,32,40)(20,25,22,27)(21,26,35,43)(29,37,31,39)(30,38,45,47)(33,41,36,44)(34,42,46,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,40)(20,31,22,29)(21,34,35,46)(25,39,27,37)(26,42,43,48)(30,36,45,33)(38,44,47,41);(9,12,13,11)(10,14,15,16)(19,29)(20,22)(24,37)(25,27)(30,34)(31,32)(33,36)(38,42)(39,40)(41,44)(45,46)(47,48);(1,3,4,7)(2,6,5,8)(17,20,23,22)(18,25,28,27)(19,29,32,31)(21,33,35,36)(24,37,40,39)(26,41,43,44)(30,34,45,46)(38,42,47,48);(2,5)(6,8)(17,21,23,35)(18,26,28,43)(19,30,32,45)(20,33,22,36)(24,38,40,47)(25,41,27,44)(29,34,31,46)(37,42,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6598067);