#6590147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497343);