#6434187

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,36)(21,27)(22,28)(23,39)(24,29)(25,34)(26,37)(31,40)(32,46)(33,42)(35,44)(38,43)(41,48)(45,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,35)(18,25,29,42)(20,31,21,32)(22,34,23,33)(26,40,43,46)(27,41,36,45)(28,44,39,30)(37,48,38,47);(11,15)(13,16)(17,20,24,21)(18,26,29,43)(19,31,35,32)(22,37,23,38)(25,40,42,46)(27,39,36,28)(30,45,44,41)(33,47,34,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,24,20)(18,27,29,36)(19,32,35,31)(22,37,23,38)(25,41,42,45)(26,39,43,28)(30,46,44,40)(33,47,34,48);(17,22,24,23)(18,28,29,39)(19,33,35,34)(20,37,21,38)(25,30,42,44)(26,27,43,36)(31,47,32,48)(40,45,46,41)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341383);