#6590436

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 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)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,25)(20,23)(21,26)(22,34)(24,27)(28,31)(29,40)(30,32)(33,36)(35,39)(37,41)(38,42)(43,44)(45,47)(46,48);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,32)(21,29)(22,31)(23,30)(24,38)(26,40)(27,42)(28,34)(33,47)(35,48)(36,45)(37,44)(39,46)(41,43);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,23,18)(19,28,30,42)(21,35,36,41)(22,34,24,27)(25,38,32,31)(26,37,33,39)(29,40,45,47)(43,44,48,46);(9,12,13,11)(10,16,15,14)(17,21,23,36)(18,26,20,33)(19,29,30,45)(22,37,24,39)(25,40,32,47)(27,35,34,41)(28,43,42,48)(31,44,38,46);(1,4,5,3)(2,6,8,7)(17,22)(18,27)(20,34)(23,24)(25,32)(26,33)(29,44)(31,38)(37,39)(40,48)(43,47)(45,46)

Polycyclic presentation:

2^9|b^2=fgi,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,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, 6497632);