#6529470

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6436666);