#6543771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6450967);