#6534610

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(17,18,23,27)(19,28,30,40)(20,32,34,41)(21,26,22,35)(24,36,39,31)(25,37,42,33)(29,38,44,47)(43,45,48,46);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,27,39)(20,29,34,44)(21,31,22,36)(25,38,42,47)(26,40,35,28)(32,43,41,48)(33,45,37,46);(10,13)(14,16)(17,20,23,34)(18,25,27,42)(19,29,30,44)(21,33,22,37)(24,38,39,47)(26,41,35,32)(28,43,40,48)(31,45,36,46);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,39)(28,40)(29,44)(31,36)(38,47)(43,48)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,27,35)(19,31,30,36)(20,33,34,37)(24,40,39,28)(25,41,42,32)(29,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fgi,e^2=i,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,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6441806);